nonlinear constrained and saturated control of power...
TRANSCRIPT
AAllmmaa MMaatteerr SSttuuddiioorruumm –– UUnniivveerrssiittyy ooff BBoollooggnnaa
PhD School in
Automatic Control Systems and Operational Research
Christian Conficoni
Nonlinear Constrained and Saturated Control of
Power Electronics and Electromechanical Systems
Phd Thesis
PhD Coordinator: Advisor: Prof. Andrea Lodi Prof. Andrea Tilli
Academic year 2012‐2013
Abstract
Power electronic converters are extensively adopted for the solution of timely issues, such
as power quality improvement in industrial plants, energy management in hybrid electrical
systems, and control of electrical generators for renewables. Beside nonlinearity, this sys-
tems are typically characterized by hard constraints on the control inputs, and sometimes
the state variables. In this respect, control laws able to handle input saturation are crucial
to formally characterize the systems stability and performance properties. From a prac-
tical viewpoint, a proper saturation management allows to extend the systems transient
and steady-state operating ranges, improving their reliability and availability.
The main topic of this thesis concern saturated control methodologies, based on mod-
ern approaches, applied to power electronics and electromechanical systems. The pur-
sued objective is to provide formal results under any saturation scenario, overcoming the
drawbacks of the classic solution commonly applied to cope with saturation of power con-
verters, and enhancing performance. For this purpose two main approaches are exploited
and extended to deal with power electronic applications: modern anti-windup strategies,
providing formal results and systematic design rules for the anti-windup compensator, de-
voted to handle control saturation, and “one step” saturated feedback design techniques,
relying on a suitable characterization of the saturation nonlinearity and less conservative
extensions of standard absolute stability theory results.
The first part of the thesis is devoted to present and develop a novel general anti-windup
scheme, which is then specifically applied to a class of power converters adopted for power
quality enhancement in industrial plants. In the second part a polytopic differential in-
clusion representation of saturation nonlinearity is presented and extended to deal with a
class of multiple input power converters, used to manage hybrid electrical energy sources.
The third part regards adaptive observers design for robust estimation of the parameters
required for high performance control of power systems.
ii
Acknowledgements
At the end of these three years, I realize how much help, support and encouragement I
received from so many people. First of all I would like to thank my parents; without their
unconditioned moral (and economic) support, I would have never been able to accomplish
my studies. A special thanks to my beloved twin sister Elisa, who has been always close
to me, and a perfect neighbor. I’d like to thank all my family members in general, who
always seem so proud of me.
Much part of this thesis work has been inspired by the advices and the insightful consider-
ations of my advisor prof. Andrea Tilli. I owe him my cultural growth during these years,
both form a methodological and engineering viewpoint. Therefore I would like to thank
him for the opportunity, and for his valuable help.
My appreciation goes to prof. T. Hu who hosted me for six months at the University of
Massachusetts, Lowell, giving me the possibility to work at timely interesting topics, along
with is research team, formed by skilled and kind guys I’d like to thank too.
Finally a great thanks to all my close friends, that I know since my college, or high school
years. It is always a pleasure to hang out with such nice and smart people, often their
accomplishment motivated me to try my best during these research period. Obviously a
special though goes also to my “fellow travelers”, that is my PhD colleagues Giovanni,
Matteo, and Raffaele, the dialogue with them made this experience even more profitable.
Contents
Introduction vii
I Anti-windup Solutions 1
1 Modern Anti-Windup Strategies 2
1.1 Modern anti-windup problem statement and objectives . . . . . . . . . . . . 2
1.2 Direct Linear Anti-windup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Model Recovery Anti-windup . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Command Governor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Novel Anti-Windup strategy: general idea . . . . . . . . . . . . . . . . . . . 22
2 Saturated Nonlinear Current Control of Shunt Active Filters 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.1 SAF saturated control strategy motivation . . . . . . . . . . . . . . 28
2.2 System model and control objectives . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 State space model derivation . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Control objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Internal model-based current controller . . . . . . . . . . . . . . . . . . . . . 34
2.4 Input saturation and current bound issues . . . . . . . . . . . . . . . . . . . 35
2.5 Current control Anti-Windup scheme . . . . . . . . . . . . . . . . . . . . . . 39
2.5.1 Improvements in the anti-windup strategy . . . . . . . . . . . . . . . 44
2.6 Dealing with current and anti-windup unit limitations . . . . . . . . . . . . 46
2.6.1 General problem formulation . . . . . . . . . . . . . . . . . . . . . . 47
2.6.2 Reduced problem formulation . . . . . . . . . . . . . . . . . . . . . . 48
2.7 Current saturation strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.8 Numerical and simulation results . . . . . . . . . . . . . . . . . . . . . . . . 55
2.8.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.8.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.9 Alternative SAF AW unit design . . . . . . . . . . . . . . . . . . . . . . . . 61
3 On the control of DC-link voltage in Shunt Active Filters 64
3.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
iv
3.2 Control oriented DC-Bus capacitor sizing . . . . . . . . . . . . . . . . . . . 66
3.3 Robust integral control of voltage dynamics . . . . . . . . . . . . . . . . . . 67
3.4 Averaged control solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
II Explicit Saturated Control Design 75
4 Control of Linear Saturated Systems 76
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Reducing conservatism in saturation nonlinearity characterization . . . . . . 77
4.3 Saturated control design via LMI constrained optimization techniques . . . 80
4.3.1 Domain of attraction maximization . . . . . . . . . . . . . . . . . . . 80
4.3.2 Disturbance rejection with guaranteed stability region . . . . . . . . 82
4.3.3 Convergence rate maximization . . . . . . . . . . . . . . . . . . . . . 85
4.4 Improvements via non-quadratic Lyapunov functions . . . . . . . . . . . . . 88
4.4.1 Piecewise quadratic Lyapunov functions . . . . . . . . . . . . . . . . 89
4.4.2 Polyhedral Lyapunov functions . . . . . . . . . . . . . . . . . . . . . 91
4.4.3 Composite quadratic Lyapunov functions . . . . . . . . . . . . . . . 94
5 Control Design for Power Converters fed by Hybrid Energy Sources 99
5.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 State-space averaged model . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.1 State space description for 4 operational modes . . . . . . . . . . . . 102
5.2.2 Averaged model for the open loop system . . . . . . . . . . . . . . . 103
5.3 Saturated controller design for robust output tracking . . . . . . . . . . . . 106
5.3.1 Converting the tracking problem to a stabilization problem . . . . . 107
5.3.2 State feedback law design via LMI optimization . . . . . . . . . . . . 108
5.3.3 Numerical result for an experimental setup . . . . . . . . . . . . . . 110
5.4 Stability and tracking domain analysis via piecewise quadratic Lyapunov
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4.1 Piecewice LDI description . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4.2 Piecewise quadratic Lyapunov function . . . . . . . . . . . . . . . . 115
5.4.3 Invariance and set inclusion LMI conditions . . . . . . . . . . . . . . 117
5.4.4 Stability region estimation via LMI optimization . . . . . . . . . . . 118
5.5 Simulation and experimental results . . . . . . . . . . . . . . . . . . . . . . 119
6 Saturated Speed Control of Medium Power Wind Turbines 123
6.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 System Modeling and Control Objective . . . . . . . . . . . . . . . . . . . . 125
6.2.1 Considerations on the general control strategy . . . . . . . . . . . . 126
6.2.2 Torque and Power saturation . . . . . . . . . . . . . . . . . . . . . . 127
6.3 Saturated Speed Controller Design . . . . . . . . . . . . . . . . . . . . . . . 128
6.3.1 Controller definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
v
6.3.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.3.3 Numerical results for a case of study . . . . . . . . . . . . . . . . . . 133
6.4 Combination with MPPT approaches for speed reference generation . . . . 134
6.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
III Adaptive Nonlinear Estimation of Power Electronic and Elec-
tromechanical Systems Parameters 139
7 Polar Coordinates Observer for Robust line grid parameters estimation
under unbalanced conditions 140
7.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2 Line voltage model and adaptive estimation problem statement . . . . . . . 142
7.3 Standard adaptive observer in a two-phase stationary reference frame . . . 143
7.3.1 Simulation results and sensitivity considerations . . . . . . . . . . . 144
7.4 Nonlinear adaptive observer in a synchronous polar coordinates reference
frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.4.1 Gains selection, sensitivity analysis and simulations . . . . . . . . . 148
8 Polar Coordinate Observer for Position and Speed estimation in Perma-
nent Magnet Synchronous Machines 151
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2 PMSM Model and estimation problem definition . . . . . . . . . . . . . . . 153
8.3 Nonlinear adaptive observer based on a synchronous reference frame . . . . 154
8.3.1 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.4 Tuning rules for the proposed solution . . . . . . . . . . . . . . . . . . . . . 157
8.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
A Mathematical Tools 162
A.1 LMI feasibility problem, Eigenvalue Problem and Generalized Eigenvalue
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.2 The ellipsoid method and interior point methods . . . . . . . . . . . . . . . 164
A.3 Schur complement and S-Procedure . . . . . . . . . . . . . . . . . . . . . . . 166
A.4 Finsler’s and Elimination Lemma . . . . . . . . . . . . . . . . . . . . . . . . 168
A.5 Sector characterization for saturation and deadzone nonlinearities . . . . . . 168
B Some Considerations on Practical PI Anti-Windup Solutions 171
vi
Introduction
Control input constraints is an ubiquitus issue in control systems, even when the engineer-
ing plants are characterized by open-loop linear models, closed-loop system nonlinearity
stems form saturation at the control inputs which, in turn, is owed to the physical limita-
tions of the actuators used to apply the control effort to the plant. Nevertheless actuators
are commonly sized in order to prevent saturation under a set of nominal working condi-
tions, unpredicted phenomena such as faults, external disturbances, unfeasible references
generation, can steer the system to operate outside from the predefined working region.
Under these unexpected conditions the required control effort can act beyond the admis-
sible values, bringing the controller’s actuators to hit their limits.
It’s well known that, if not suitably handled, control input saturation can dramatically
affects the feedback system, giving rise to the so-called windup effect. The term originates
from the first simple PID controllers, implemented by means of analog electronics, where
the actuator saturation slows down the system response, causing the integral part of the
controller to windup to large values and, as a consequence, long settling time and excessive
overshoot. The term is still used in more involved multivariable modern controllers, to
denote a pretty severe performance degradation of the system under saturation, which, in
some particular cases, could also lead to the loss of the system stability properties.
From the early 40’s were practictioners became aware of saturation issues until a not so
distant past (70’s and early 80’s), and still in several industrial applications, the prob-
lem of input saturation was handled by means of ad hoc solutions; namely the controller
was designed disregarding constraints, then the system was augmented with application-
specific schemes, whose task was to introduce additional feedbacks, in such a way that the
overall system had a graceful behavior also under saturation. For this reason, this kind
of additional systems were referred to as anti-windup units. Even though this schemes
were able to effectively deal with saturation of the specific plant, a formal stability and
performance characterization was lacking.
The first academic attempts to rigorously cope with constrained control inputs, providing
constructive techniques for a broader class of systems, were made in the early 80’s, first
by working on intelligent integrators [1], [2] then, in the late 80’s early 90’s introducing
state space interpretations [3], [4] [5], [6] valid also for multivariable systems. However, at
this point, most of the approaches still were not handling stability and performance in a
systematic fashion. Moreover methodological tools to tune the anti-windup system were
not in general provided. It’s only in the last two decades that so-called modern approaches
vii
were proposed to cope with input saturation. A settled definition of modern approach to
input saturation is still missing, however, in most of the dedicated literature, the term is
referred to methodological and systematic approaches, based on sound theoretical results,
that can be exploited for a rather general class of systems, in order to formally and quan-
titavely ensure stability and a certain degree of performance under saturation.
This definition clearly goes beyond the solely anti-windup compensators; in this respect
three main class of methodologies can be outlined
• Modern Anti-windup compensator design
• Direct saturated feedback control design
• Model predictive control
Modern anti-windup compensators are the natural continuation of the above mentioned
applied perspective driven trend, began in industries since the first stage of control the-
ory. Similarly to their “ancestors” this schemes are joined to preexisting controllers, that
have been designed to meet the specifications without explicitly considering possible con-
straints, with the specific task to tackle saturation by modifying the original controller
output signals and states or, for certain approaches, the closed-loop system reference ([7],
[8]). Differently from early compensators, recently proposed anti-windup units can pro-
vide stability and performance guarantees. Furthermore the design can be carried out
by means of reliable synthesis algorithms, based on well established mathematical tools,
e.g. Linear Matrix Inequalities (LMI) [9]. From a historical standpoint, these techniques
started to be developed in the late 90’s ([10], [11]), nowadays the topic is quite mature,
and recent publications ([12], [13], [14]) attempted to order the vast amount of contribu-
tions in a comprehensive an self-contained way. However it still remain an open research
issue, especially for what concerns solutions for nonlinear constrained systems with non
minimum phase zero dynamics, and complex systems whose specifications are hard to be
cast into the available modern anti-windup frameworks.
An alternative approach to deal with saturation is to design the control law in one step,
i.e. taking explicitly into account input constraints during the regulator design procedure.
While anti-windup comes from practical industrial problems, this approach stemmed from
the works on Lyapunov absolute stability theory. In plain words, the main idea is to
characterize the saturation nonlinearity in a less conservative with respect to the classic
sector characterization adopted to derive Popov and Circle criteria ([15]). New results on
this topic began to be developed from the 80’s onward, mainly searching for possibly local,
but tighter sector conditions for the saturation nonlinearity standard sector characteriza-
tion ([16]), and then handle the saturated system analysis and design as a standard Lure
problem. Another possible methodology consists in giving a (possibly local) description
of the saturated closed-loop system in terms of parametrized Polytopic Linear Differential
Inclusions (PLDIs) (see [17], [18], [19]), commonly adopted in the robust control theory
framework. Similar approaches were proposed also for deadzone function and more general
classes of algebraic nonlinearities [20]. With this representation at hands, both analysis
viii
and control synthesis problem are addressed by applying the Lyapunov’s second method.
The main advantage of this approach is that control law synthesis can be cast into nu-
merically efficient optimization algorithms involving linear matrix inequalities, where the
objective function is selected to formally meet the specifications ([21], [22]). Moreover, in
the recent past, several solutions, based on non-quadratic Lyapunov functions, combined
with nonlinear feedback controllers, have been proposed, exploiting the similarity with
typical system descriptions addressed in the robust control theory framework ([23], [24],
[25]). This has led to further reduce and eliminated ([26])conservatism in the LDI anal-
ysis and stabilization procedure (the approximation in the nonlinearity description still
hold), on the other hand, this has come “hand in hand” with an increased complexity
in the analysis and design algorithms. Another limitation is that the majority of these
works deals with saturated linear plants, while one step constrained control of nonlinear
systems is still an open problem. It’s further to notice that this two class of approaches
are somewhat interlaced, since some of the the early results on analysis and synthesis of
saturated linear systems have been exploited to produce the above mentioned LMI-based
tuning algorithms for modern anti-windup units.
The third approach to handle input saturation is by casting the problem in the Model
Predictive Control (MPC) framework; indeed saturated systems are a particular class of
constrained systems. MPC for input saturated systems was proposed since the late 80’s
and early 90’s ([27] [28]), and nowadays, thanks to the advance in both theoretical ([29])
and technological fields, it can be exploited to handle rather complex nonlinear saturated
systems. Differently from the previous solutions, MPC is more suitable to be adopted when
the plant is expected to hit the saturation limits commonly also during its nominal opera-
tion. When saturation is expected not to occur so frequently, or when a control structure
has been already implemented, the other two approaches seems preferable. Furthermore,
in some cases it can result hard to cast the systems specifications into the standard MPC
framework, even for what concerns it’s desired “small signals” behavior, namely the range
of states and inputs for which no constraints are active.
For the class of power electronic and electromechanical systems, the issue of input satu-
ration is of crucial importance; since, in addition to the above mentioned problems, such
devices often are required to manage high energy/power levels, and power converters are
directly connected to the line grid. Therefore they can be characterized as safety criti-
cal components, whose behavior has to be preserved under a wide range of possible non
nominal scenarios. Furthermore, from an engineering point of view, an optimal use of
the actuators capacity, would avoid system oversizing, reducing the overall system costs,
mass, and volume. Finally, the extension of the system operational regions, provided by
a correct saturation management, enhances the system reliability and availability proper-
ties. Recently this topics have been receiving particular attention, due to the technological
advances, that made power electronic components a pervasive presence in several indus-
trial and commercial fields. In particular power electronics is extensively adopted in the
recently blossomed fields of optimal electrical energy management, power quality enhance-
ix
ment, and renewable energies. It’s further to remark that these kind of complex dynamic
devices, are expected to distributed into complex, possibly “smart”, grid topologies, for
which more demanding regulations, compared to traditional networks, have to be fulfilled.
From a control theory viewpoint, the state-space models of converters and electric ma-
chines adopted in the above mentioned power systems, are commonly nonlinear, and,
in some cases, the controlled outputs are characterized by non-minimum phase internal
dynamics, usually related to energy reservoirs as DC-link capacitors and magnetic fields
in electric machines. These features make high-performance control quite complex, even
under saturation-free hypothesis, therefore system constraints need to be addressed by
means of modern, sophisticated solutions.
Thesis Outline and Contributions
This thesis is devoted to present and discuss advanced saturated control techniques for
power electronic and electromechanical systems. Following the spirit of the modern ap-
proaches previously outlined, solutions capable to tackle saturation in a systematic and
formal fashion are proposed, overcoming the drawbacks of classic countermeasures usually
adopted in industries. In fact, standard techniques are focused on preventing system sat-
uration, and extending the system working range, considering some steady-state system
configurations. The main aim of this work is to move forward; ensuring a correct sat-
uration handling even during transient conditions, and for rather complex scenarios not
covered before. In order to achieve this objective, the starting point is to construct rather
general advanced methodologies, inspired by the existing modern methodologies, then ex-
ploit the guidelines given by this general approaches to design specific saturated control
laws for the considered power electronics application. This thesis gathers the research
work carried out in the last three years, and, for sake of clarity, is divided into three main
parts.
The first part (chapters 1 − 3) concerns anti-windup unit design issues, with specific ap-
plications to power converters devoted to power quality enhancement in industrial plants.
In chapter 1 first the anti-windup design problem and its objectives are formulated ei-
ther from a qualitative and a quantitative standpoint, then the three most main modern
anti-windup strategies proposed in the literature are described in their basic features. Fi-
nally a novel anti-windup approach, first introduced in [30], is defined. The general idea
is described and then formally developed for a simple but rather general class of control
systems, providing an interpretation in the light of the modern anti-windup schemes ob-
jectives, and critically comparing the presented methodology with the other approaches
proposed in the literature.
In chapter 2 this strategy is exploited to design a constrained control solution for a class of
power converters used as active filters for reactive and harmonic currents compensation in
industrial plant. The initial part of the chapter is devoted to describe the system, define
the control objectives, and present a robust saturation-free current control solution [31],
x
[32] that will be used as a benchmark to test the proposed anti-windup strategy. Then
it’s shown how the general anti-windup scheme can be adapted to manage the current
control unit saturation. The basic strategy is extended with some specific countermea-
sures, applied to improve the filter performance and, at the same time, deal with the plant
non minimum phase zero-dynamics, associated with the voltage of the converter DC-link
capacitor. Finally the anti-windup unit is suitably combined with a current saturation
strategy, to formally handle the converter current limitations, that, from a control theory
standpoint, can be regarded as plant state constraints.
For the sake of completeness, and to clarify it’s possible interaction with the anti-windup
unit, Chapter 3 is devoted to the issue of the DC-link voltage dynamics stabilization. After
a formal definition of the control problem, some possible control strategies are discussed.
Among them, the focus will be put on a promising averaging control solution [33], based
on the two-time scale separation of the filter current and DC-link voltage dynamics.
The second part (chapters 4 − 6) regards one step control design solutions for input sat-
urated linear and bilinear systems, also in this case with applications to timely power
electronic issues. In chapter 4 the theoretical background, that will be exploited and the
extended to cope with the considered power systems, is presented; the above mentioned
saturation characterization, based on parametrized polytopic differential inclusions, is an-
alyzed and formally motivated, showing how conservatism is reduced with respect to stan-
dard characterization techniques. Then the controller synthesis procedure is presented,
first by considering linear state feedback laws and quadratic Lyapunov candidates, that
lead to the formulation of LMI constrained convex problems, allowing to select the optimal
saturated feedback controller to satisfy common requirements such as stability region max-
imization, disturbance rejection, or convergence rate enhancement. Then some common
classes of non-quadratic Lyapunov functions, such as piecewise quadratic [34], polyhedral
[35] or the so-called composite quadratic functions [36], possibly combined with nonlinear
feedback controllers, aimed to reduce conservatism in the design procedure, and improve
the results obtainable with quadratic functions and linear controllers, are presented.
In chapter 5 this methodologies are adopted and extended, according to the procedure
proposed in [37], to design saturated control laws for a class of multiple inputs power elec-
tronic, converter extensively adopted in a wide range of applications (e.g hybrid electrical
vehicles and stand-alone photovoltaic systems) to actively steer the power flow between hy-
brid electrical energy storage devices, such as supercapacitors and batteries, and a generic
load. Beside the hard input constraints, this systems are also typically characterized by bi-
linear state-space models, hence PLDI-based description of saturated systems is extended
to describe also bi-linearity, so that the obtained inclusion is ensured to contain all the
trajectories of the original nonlinear system. Then a robust control solution is designed
by means of an LMI constrained convex problem. Finally the tracking domain analysis
is improved by means of a particular class of piecewise quadratic functions based on a
suitable partition of the state space region.
Chapter 6 deals with control solutions for medium power wind energy conversion systems
xi
[38], it goes a little astray from the path of the part, as the above mentioned saturated
control techniques are not applied. However, the power/torque saturation limits of the
generators are handled by a specific strategy, presented in [39], that consists in combining
the two system’s control knobs, i.e. the generator torque and the blade pitch angles, to
extend the system operating region for wind speeds producing a power/torque that goes
beyond the turbine limits, and to prevent windup effects or control signal bumps when the
nominal conditions are restored. Since the system limits are explicitly accounted during
the control law design and no additional units with the specific task to handle saturation
are needed, this can be somewhat seen as a one step saturated control synthesis procedure.
The final part of this thesis (chapters 7− 8) deals with auxiliary tools for power electronic
and electromechanical system control. Very frequently, advanced control units for this
class of systems require accurate informations about state variables or parameters that
are not measured, or whose measures can be impaired by the harsh conditions under which
the system has to operate. Moreover, in many cases it can be preferable to reduce the
number of sensors placed on the plant, for cost saving and improved reliability reasons.
Hence such variables need to be robustly estimated exploiting the informations on the
system model and the other measured variables. A classic example in the considered field
regards sensorless control algorithms for electric machines, where the position and speed
of the rotor are reconstructed without the need of any sensor. From a control theory view-
point the most widely adopted frameworks to cope with this issues are adaptive observers
designed by means of passivity arguments [40] or nonlinear internal model principles [41],
and high gain observers [42].
In this context in chapter 7 a nonlinear adaptive observer, first introduced in [43], de-
voted to robust three-phase line grid parameters estimation, under possible unbalanced
conditions, is discussed. The basic idea is to exploit the properties of a polar coordinates
synchronous reference frame to represent the line voltage, in order to provide a larger
robustness to withstand negative sequences and voltage harmonics (produced by line un-
balancing) with respect to standard LTI observers.
The same principle is adopted in chapter 8 for rotor speed and position reconstruction
for Permanent Magnet Synchronous Machines (PMSM) without the assumption of stator
flux dynamics perfect knowledge. The main idea is to push the reference frame adopted
in the observer toward the synchronous one, by forcing it to be intrinsically aligned with
the estimated back-emf vector, and by designing suitable adaptations law for its speed
and angle along with the back-emf amplitude. Since the flux dynamics are not integrated
for the estimates, an increased robustness with respect to measurements uncertainties is
obtained. Following the approach of [44], the adaptation law is tuned exploiting singular
perturbation theory arguments and Lyapunov’s second method based design techniques,
in order to formally ensure the the observer’s estimates asymptotic convergence. Two
appendices conclude this work; in A the mathematical tools, used throughout the thesis
to formulate and solve LMI constrained optimization problems for anti-windup or con-
strained feedback design purposes, are recalled, while in B some practical considerations
xii
about anti-windup solutions for standard proportional integral regulators under non sym-
metric or time-varying saturation boundaries are reported.
xiii
xiv
Part I
Anti-windup Solutions
1
Chapter 1
Modern Anti-Windup Strategies
This chapter reviews the state-of-the-art in the field of modern anti-windup
methodologies. The anti-windup unit objectives are formally stated and moti-
vated. Then the three most widely used families of modern anti-windup ap-
proaches are presented, finally a novel anti-windup strategy is discussed in its
main features, considering a simple but rather general class of nonlinear sys-
tems.
1.1 Modern anti-windup problem statement and objectives
Consider the generic input saturated feedback control systems reported in Fig. 1.1(a),
where w(t) is an exogenous input, y(t) and z(t) are the the measured and performance
outputs respectively, while u is the control input vector that enters the actuator, repre-
sented by its nonlinear saturation function, before acting on the plant. For the sake of
simplicity the considered actuator nonlinearity is described as a decentralized symmetric
saturation function, i.e. a vector of scalar saturation functions, as the one reported in Fig.
1.1(b), where the ith function depends only on the ith input component, an the satura-
tion limits are symmetric with respect to the origin (u ∈ [−usat, usat]), according to the
following law
sat(ui) = sign(ui)minusat, |ui|), i = 1, . . . ,m (1.1)
however the anti-windup schemes presented in this chapter are valid for more general
classes of memoryless nonlinearities. Here and in the following, with some abuse of nota-
tion, the symbol sat(·) will be used to denote either the vector and the scalar saturation
functions. As mentioned in the Introduction, windup phenomena can take place when the
control input saturation occurs, since the unconstrained controller is designed neglecting
the system constraints. The main idea of anti-windup solutions is to augment the system
with a specific unit that prevent the controller to misbehave during saturation. In this
respect, from a qualitative , the following objectives can be defined
• Small signals preservation: It is the first objective of any anti-windup scheme, it con-
sists of making the responses of the augmented system equal to those of a saturation-
2
1.1. Modern anti-windup problem statement and objectives
Plant
Unconstrained
Controller
w
sat(u)y
u
(a) Saturated feedback scheme.
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
u
sat(u)
(b) Saturation nonlinearity
Figure 1.1: Saturated feedback system.
free system, with no anti-windup augmentation, unless saturation occurs. In other
words, it is required the anti-windup system to be silent when the required control
effort lies in the actuators admissible range, according to a reasonable “parsimony
principle”;
• Internal stability : Considering the absence of exogenous inputs, make a defined
working point, that without loss of generality is assumed to be the origin, asymptot-
ically stable, maximizing the basin of attraction or, at least ensure a stability region
containing the set of states over which the system is expected to operate. It’s fur-
ther to notice that this requirement is more ambitious then simple local asymptotic
stability, that would be guaranteed also by any stabilizing unconstrained controller.
Indeed for plants that are not exponentially unstable, global asymptotic stability
can be induced by means of suitable anti-windup solutions;
• External stability : To enforce a bounded response for the set of initial conditions
and exogenous inputs that are expected during operation. A bounded response
for any initial condition and exogenous input is impossible to achieve for certain
classes of constrained systems, e.g linear plants with exponentially unstable modes.
Thus it’s not reasonable to generally express external stability in global terms, such
requirement can be asked for linear exponentially stable systems. On the other hand
for this class of systems providing stability via anti-windup augmentation is trivial,
hence further objectives are commonly pursued as those discussed in the next item;
• Unconstrained response recovery : To reproduce the closed-loop response of a virtu-
ally saturation-free system whenever possible. It looks reasonable to require that,
beside the linear actuator region, the augmented closed loop system emulates the
saturation free system behavior also when input saturation has occurred, if possible.
When this is not the case, the internal and external stability objectives become rele-
vant. However, it’s further to remark that characterizing the unconstrained response
recovery property can be tricky. While for exponentially stable plants it’s easy to
3
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
Plant
Unconstrained
Controller
w
sat(u) y
u
w
z
An -windup
Unit
+
-
q
v
Figure 1.2: Input saturated system with anti-windup augmentation.
verify that the property is achievable if the unconstrained input uuc(t) asymptoti-
cally falls below the saturation threshold, i.e. uuc(t) − sat(uuc(t)) → 0 if t → ∞,
for exponentially unstable or marginally stable plants this condition is no longer
sufficient (see [12] ch. 2 for further details).
The above qualitative tasks can be mapped into quantitative performance indexes, on
which modern techniques relies, to define the anti-windup design algorithms. As regards
the small signal preservation property, it can be structurally enforced by means of the
standard adopted anti-windup architecture, reported in Fig. 1.2; since the anti-windup
system is fed by the mismatch signal q = sat(u)− u between the controller signal and its
saturated version, it suffices to design a generic anti-windup unit
xaw = f(xaw, q)
v = h(xaw, q)(1.2)
satisfying f(0, 0) = 0, h(0, 0) = 0 to guarantee small signal preservation.
As regards internal and external stability, a reasonable choice is to relate both the
properties to the quantity
‖z‖2 ≤ β|xcl(0)|+ γw(t) (1.3)
with β, γ two positive scalars and xcl = [xp xc xaw]T ∈ R
np×nc×naw the augmented closed-
loop state collecting the plant xp, controller xc and anti-windup unit xaw states. While
‖ · ‖2 denotes the L2 norm defined as√
(∫ t0 x(τ)
Tx(τ)dτ), differently, along the thesis the
euclidean norm x(t)Tx(t) will be denoted with the symbol ‖‖.The L2 size of z(t) is a standard performance indexes for control systems with exogenous
inputs, and it relies on the assumption that the size of the plant state is related to that
of the performance output . The anti-windup synthesis objective can then be formulated
as to minimize the L2 gain γ from w to z. Unless the plant is exponentially stable,
it’s impossible to ensure a global finite L2 whatever the anti-windup augmentation is.
However, it is always possible to enforce a finite gain over a local region, the aim in this
case is to maximize the finite gain region while still minimizing γ. In general these are two
contrasting objectives, as it turns out that, the wider the finite gain region is, the larger
4
1.1. Modern anti-windup problem statement and objectives
the upper bound on the L2 gain. In this respect, it’s has been remarked [12] that using
a single factor to characterize the energy attenuation (or amplification) in a constrained
feedback loop can be misleading, since a saturated feedback controller able to attenuate
a low energy exogenous input, may not be able to attenuate an high energy input by the
same factor. Therefore it makes more sense to characterize the energy attenuation of the
exogenous input w through a nonlinear L2 gain function γ(s) where s is the exogenous
input energy level. There are basically three kind of shapes that the nonlinear gain can
assume; if the system is both open and closed-loop globally exponentially stable, then the
nonlinear gain can be upper bounded by a linear function, however, for low energy input
levels, the ratio between the output energy and the input energy can be much smaller
than the linear bound. If the closed-loop system is externally stable for low energy inputs
but not for high energy inputs, we have a gain function that grows to infinite for finite
energy exogenous inputs, this is typical of open loop unstable plants, locally stabilized by
a constrained feedback. Finally we can have a nonlinear curve that does not go to infinite
for limited value of ‖w‖2, but it cannot even be bounded by a linear function. In this case
the system is said to be L2 stable, but without a global finite gain γ, i.e. the output energy
may grow unbounded if the energy of the input increases. This situation typically occurs
for marginally stable plants globally asymptotically stabilized by a constrained feedback
law.
As regards the unconstrained response recovery, a reasonable assumption is the mismatch
between the unconstrained and the augmented system performance outputs zuc − z is
somewhat related to the mismatch of the corresponding plant states xpuc−xp. Then if z−zuc belongs to the class of L2 signals and it’s uniformly continuous, according to Barbalat’s
lemma [15] we can conclude that the outputs difference asymptotically goes to zero, and
so the state mismatch. Hence, a way to guarantee the unconstrained response recovery
property, for inputs w(t) that make the unconstrained control input uuc to converge, in a
L2 sense, to the linear region of the saturation function, is to fulfill the following inequality
‖z − zuc‖2 ≤ β|xaw(0)|+ γ|uuc − sat(uuc)| (1.4)
for the set of initial conditions and unconstrained control inputs that are expected dur-
ing the system operation. Continuous uniformity of the considered signals is satisfied by
anti-windup augmented system under mild technical hypothesis (see [12] ch. 2). Also
in this case the anti-windup design objective is to minimize γ, and the optimal valued
is commonly called unconstrained response recovery gain. Even if focused on the perfor-
mance output recovery, this objective is similar to the standard L2 formulation (1.3) for
the augmented system stability, hence similar considerations about locally valid finite gain
and nonlinear gain functions can be made.
Before analyzing the different classes of anti-windup algorithms, we conclude this para-
graph with some standard nomenclature adopted to subdivide the different anti-windup
schemes:
• Static or Dynamic: When the anti-windup unit can be represented as a memoryless
5
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
function the scheme is refereed to as static, otherwise it is said to be dynamic. In the
literature there are two different definitions of a full order dynamic scheme, some
authors [45] mean the anti-windup unit order is equal to the plant order i.e. np =
naw, while others [13] include the controller dynamics in the definition, considering
as full order a compensator whose order is equal to the closed loop system order
before augmentation, i.e. naw = np + nc;
• Linear or non linear : In the case the original system is augmented with linear
dynamics, the anti-windup scheme is said to be linear, nonlinear otherwise;
• External or full authority : In all the modern anti-windup solutions that can be rep-
resented by the scheme in Fig. 1.2, the anti-windup signals v enter the unconstrained
controller additively. In some realizations the anti-windup outputs can directly act
on both the controller output and state equations, these are commonly refereed
to as full authority anti-windup augmentation. For example, considering a linear
controller, a typical full authority solution corresponds to the following equations
xc = Acxc +Bcy +Bcww + v1
yc = Ccxc +Dcy +Dcww + v2.(1.5)
While for other instances the anti-windup unit injects the signals only at the input
and output of the unconstrained controller, for this reason this kind of schemes are
called external anti-windup augmentation. In this case, example (1.5) need to be
modified asxc = Acxc +Bc(y + v1) +Bcww
yc = Ccxc +Dc(y + v1) +Dcww + v2.(1.6)
1.2 Direct Linear Anti-windup
The direct linear anti-windup (DLAW) approach was the first constructive scheme relying
on LMI-based optimization problem to tune the anti-windup unit in order to formally
ensure the stability and performance properties outlined before.
First simple static schemes were considered [46], [9], more recently, extensions of the
approach involving dynamic compensators have been proposed [45], [47]. In this section
the basic results will be presented considering dynamic anti-windup schemes, since the
static schemes can be derived as particular cases. The target for this class of solutions,
are linear saturated plants in the form
xp = Apxp +Bpusat(u) +Bpww
y = Cpxp +Dpusat(u) +Dpww
z = Czxp ++Dzusat(u) +Dzww
(1.7)
associated with a linear dynamic controller
xc = Acxc +Bcuc +Bcww + v1
yc = Ccxc +Dcuuc +Dcww + v2(1.8)
6
1.2. Direct Linear Anti-windup
where uc ∈ Rp is the controller input yc ∈ R
m the output and v1, v2 two anti-windup
signals. A natural assumption is that the above controller provides stability and the
required performances of the closed-loop system when no saturation occurs, i.e. the system
obtained by the saturation-free interconnection: sat(u) = u = yc, uc = y, v1 = v2 = 0,
is globally asymptotically stable. In other words, assuming also well-posedness of the
interconnection, i.e. the matrix ∆ = I −DcDpu is non singular, the controlled is required
to make the unconstrained system closed-loop state matrix Hurwitz, that is
A =
[
Ap +Bpu∆−1DcCp Bpu∆
−1Cc
Bc(I −Dpu∆−1Dc)Cp Ac +BcDpu∆
−1Cc
]
< 0. (1.9)
In DLAW approaches the compensator in Fig. 1.2 is selected as the following linear filter,
producing the signal v = [v1 v2]
xaw = Aawxaw +Baw(sat(u)− u)
v1 = Caw,1xaw +Daw,1(sat(u)− u)
v2 = Caw,2xaw +Daw,2(sat(u)− u).
(1.10)
In general the order naw of the filer is a design parameter along with the matrices Aaw,
Baw, Caw = [CTaw,1C
Taw,2]
T , Daw = [DTaw,1D
Taw,2]
T . For the sake of brevity, here only
full order schemes (naw = nc + np) with some reference to static versions (naw = 0)
as particular cases, are recalled. The most common performance measure, optimized by
DLAW strategies, is the input-output gain form w to z that, by (1.3), can be expressed
as the inequality ‖z‖2 ≤ γ‖w‖2.Considering the usual interconnection u = yc, uc = y, and expressing the saturation
nonlinearity in terms of the mismatch signal q, i.e. sat(yc) = yc + q, the systems (1.7),
(1.8), (1.10) can be combined to obtain the following augmented closed-loop system
xcl = Aclxcl +B1clq +B2clw
ycl = C1clxcl +D11clq +D12clw
z = C2clxcl +D21clq +D22clw
(1.11)
with
Acl =
[
A BvCaw
0 Aaw
]
, B1cl =
[
Bq +BvBaw
Baw
]
, B2cl =
[
B2
0
]
C1cl =[
C1 Cv1Caw
]
, D11cl = D1 + Cv1Daw, D21cl =D2 + Cv2Daw
C2cl =[
C2 Cv2Caw
]
7
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
u
sat(u)dz(u)
Figure 1.3: Deadzone nonlinearity corresponding to the mismatch signal q.
and
Bv =
[
Bpu∆−1[0 Im]
BcDpu∆−1[0 Im] + [Inc 0]
]
, Bq =
[
Bpu(Im +∆−1DcDpu)
BcDpu(Im +∆−1DcDpu)
]
C1 =[
∆−1DcCp ∆−1Cc
]
, Cv1 = ∆−1[
0 Im
]
C2 =[
Cz +Dzu∆−1DcCp Dzu∆
−1Cc
]
, D1 = ∆−1DcDpu
Cv2 = Dzu∆−1[0 Im], D2 = Dzu(Im +∆−1DcDpu)
B2 =
[
Bpu∆−1(Dcw +DcDpw) +Bpw
BcDpu∆−1(Dcw +DcDpw) +Bcw +BcDpw
]
, D12cl = ∆−1(Dcw +DcDpw)
D22cl = Dzw +Dzu∆−1(Dcw +DcDpw)
Note that, if a decentralized symmetric saturation (1.1) is concerned, the mismatch signal
corresponds to a decentralized deadzone nonlinearity q = dz(u), i.e. each component of
the control input vector is processed according to the function reported in Fig. 1.3. In
order to develop LMI conditions for the compensator synthesis, the following generalized
sector characterization of the deadzone function, first introduced in [48], is commonly
exploited: define the set S(usat) := u ∈ Rm, ω ∈ R
m : −usat ≤ u− ω ≤ usat then the
following holds
Lemma 1.2.1 If u and ω belongs to the set S(usat), then the nonlinearity q(u) = sat(u)−u satisfies the following inequality
q(u)TS−1(q(u) + ω) ≤ 0 (1.12)
for any diagonal positive definite matrix S
the proof of the lemma is reported in A.5. Based on this characterization, in [47] is
provided the following sufficient condition for the global stability of the augmented closed-
loop system (1.11)
8
1.2. Direct Linear Anti-windup
Proposition 1.2.2 If there exist a symmetric positive matrix Q ∈ Rn×n, where n =
naw + nc + np, a diagonal matrix S ∈ Rm×m and a positive number γ such that
QATcl +AclQ B1clS −QCT
1cl B2cl QCT2cl
(B1clS −QCT1cl)
T −2(S + 2D11clS) −D12cl SDT21cl
BT2cl −DT
12cl −I DT22cl
C2clQ D21clS D22cl −γ2I
< 0 (1.13)
then
• If w = 0, the origin of system (1.11) is globally asymptotically stable;
• The closed loop system trajectories are bounded for any initial condition and any
w(t) ∈ L2;
• The system is externally L2 stable with
∫ T
0z(t)T z(t)dt ≤ γ2
∫ T
0w(t)Tw(t)dt+ γ2xcl(0)
TQ−1xcl(0), ∀ T ≥ 0. (1.14)
Proof Consider the quadratic candidate Lyapunov function V (xcl) = xTclQ−1xcl. Then
both internal and external stability are ensured for any xcl(0) if
V (xcl) < γ2wTw − zT z. (1.15)
Lemma 1.2.1 hold globally if ω = u, hence we have
qTS−1(q + u) ≤ 0.
Therefore, applying the S-procedure to the two inequalities above we obtain
V +1
γ2zT z − wTw − 2qTS−1(q + u) < 0 (1.16)
which, by Schur’s complement, is equivalent to (1.13). Then it’s easy to verify that
when w = 0, V (xcl) < 0 , hence global asymptotic stability trivially follows. Condition
(1.14) is be obtained by integrating (1.16), then Lemma 1.2.1 and positive defiteness of V
(V (xcl(T )) > 0 for T > 0) yields
∫ T
0zT zdt ≤ γ2V (xcl(0)) + γ2
∫ T
0wTwdt ≤ γ2xcl(0)
TQ−1xcl(0) + γ2∫ T
0wTwdt (1.17)
As long as the synthesis problem is faced, conditions (1.13) become not convex, in par-
ticular bilinear matrix inequalities (BMI) arise, since the products between the matrix
variables Aaw, Baw, Caw, Daw, Q and S appears in several inequality terms. However in
the case of full order compensator a convex characterization can be obtained as stated in
the next result
9
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
Proposition 1.2.3 An anti-windup compensator in the form (1.10) can be designed to
satisfy proposition 1.2.2 if there exist two positive definite symmetric matrices X,Y ∈Rnc+np × nc + np and a positive number γ such that
AX +XA XB2 CT2
BT2 X −I DT
22cl
C2 D22cl −γI
< 0
Y1ATp +ApY1 Bpw Y1C
Tz
BTpw −I DT
zw
CzYT1 Dzw −γ2I
< 0,
[
X I
I Y
]
> 0
(1.18)
where Y1 is the upper left corner square block, with dimension np, of Y .
Proof First partition the matrix Q defined in proposition 1.2.2 as
Q =
[
Y NT
N W
]
, MTN = I −XY
Q−1 =
[
X M−1
M W
] (1.19)
then define the matrices
φ1 =
Y AT + AY ANT BqS − Y TCT1 B2 Y CT
2
NAT 0 −NCT1 0 NCT
2
(BqS − Y TCT1 )
T BT2 −2S −D1S − SDT
1 −D22cl SDT2
C2Y C2NT D2S D22cl −γ2I
F =
[
0 I 0 0 0
BTv 0 −CT
v1 0 CTv2
]
G =
[
N W 0 0 0
0 0 S 0 0
]
, H =
[
Aaw Baw
Caw Daw
]
(1.20)
noting that (1.13) can be rearranged as
φ1 + F THG+GTHTF < 0 (1.21)
and using, the Elimination Lemma, the following equivalent inequalities are obtained
NTF φ1NF < 0, NT
Gφ1NG < 0 (1.22)
where NF , NG are basis of Ker(F ), Ker(G) respectively. Since NG can be defined as
NG =
X 0 0
M 0 0
0 0 0
0 I 0
0 0 I
(1.23)
10
1.2. Direct Linear Anti-windup
(1.22) is equivalent to the first inequality in (1.18). Similarly we can define
NTF =
[Inp 0] 0 Bpu 0 0
0 0 0 I 0
0 0 Dzu 0 I
(1.24)
noting that the following equalities hold
[Inp 0]Bv −BpuCv1 = 0, −DzuCv1 + Cv2 = 0
BpuSBTq
[
Inp
0
]
+ [Inp 0]BqSBTpu = 0
Dzu(−2S − 2D1S)DTzu + 2D2SD
Tzu = 0
(1.25)
it can be concluded that inequality involving NF in (1.22) is equivalent to the second
inequality in (1.18). Finally, the third inequality of (1.18) allow to define Q and therefore
X, Y satisfying (1.19).
It’s further to notice that proposition (1.2.3) does not provide a constructive method to
synthesize the anti-windup filter, such conditions can be found in the particular case of
static DLAW ([9]) where only Daw need to be computed. However, numerically tractable
synthesis algorithm for dynamic DLAW can be defined by fixing some of the variables, as
in the following example
1. Minimize γ under the LMIs in (1.18);
2. Compute Q by (1.19);
3. Fix Q in (1.13) and solve the resulting eigenvalue problem in the variables Aaw,
Baw = BawS, Caw, DawS.
It’s further to remark that global results can be ensured only for exponentially stable
plants, it’s easy to verify that if it’s not the case the previous conditions would be un-
feasible. Similar results have been established in a local context [49]; exploiting the same
framework as in proposition 1.2.2 the following yields
Proposition 1.2.4 If there exists a symmetric positive definite matrix Q ∈ Rn×n, a ma-
trix Z ∈ Rm×n, a positive diagonal matrix S ∈ R
m×m and a positive scalar γ such that
QATcl +AclQ B1clS −QCT
1cl − ZT B2cl QCT2cl
(B1clS −QCT1cl − ZT )T −2(S + 2D11clS) −D12cl SDT
21cl
BT2cl −DT
12cl −I DT22cl
C2clQ D21clS D22cl −γ2I
< 0
[
Q ZTl
Zlu2sat
δ
]
≥ 0, l = 1, . . . ,m
(1.26)
then
11
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
• If w = 0 the ellipsoid E(Q−1, δ) :=xcl : x
TclQ
−1xcl ≤ δis a domain of asymptotic
stability for the augmented system origin;
• For any w ∈ L2 such that ‖w‖22 ≤ δ and for xcl(0) = 0, the closed-loop system
trajectories are bounded in E(Q−1, δ), and the system is externally L2 stable with
∫ T
0zT zdt ≤
∫ T
0wTwdt, ∀ T ≥ 0. (1.27)
Proof Consider a quadratic Lyapunov candidate V (xcl) = xTclQ−1xcl, and ω = u +Kxcl
so that the sector conditions (1.12) reads as
qTS−1(q + u+Kxcl) ≤ 0 (1.28)
and it applies for any diagonal positive S and any x ∈ S(usat) where
S(usat) = u : −usat ≤ Kxcl ≤ usat . (1.29)
By choosing Z = KQ and from Schur’s complement it can be verified that the second
inequality in (1.26) implies Q−1
δ ≥ GTGu2sat
, i.e. E(Q−1, δ) ⊆ S(usat). Then applying S-
procedure to the stability condition (1.15) and (1.28), the following inequality holds ∀xcl ∈E(Q−1, δ)
V +1
γ2zT z − wTw − 2qTS−1(q + u+Kxcl) < 0 (1.30)
which by Shur complement is equivalent to (1.26). Internal local stability condition V <
0, ∀xcl ∈ E(Q−1, δ) immediately follows replacing w = 0 in the above inequality. While
(1.27) can be verified by integrating (1.30), setting xcl(0) = 0, and exploiting Lemma 1.2.1
and positive defiteness of V .
To remark the need of a trade-off between small L2 gain and wide stability regions men-
tioned in (1.1), consider the case when xcl(0) 6= 0 in a local DLAW context. A common
approach is to consider an ellipsoid E(Q−1, α) containing the admissible initial conditions,
then ensure all the trajectories starting in this set to be bounded by a larger ellipsoid
E(Q−1, α + δ), depending on the exogenous input energy level. Hence it’s clear how the
size of the stability region, basically related to α+ δ, the set of initial conditions, related
to α, and the disturbance bound on the tolerable disturbances, given by δ, are three con-
trasting objectives to be managed depending on the specific application. This topic will
be elaborated in 4.3.2 for what concerns explicit saturated state feedback design.
Similarly to the global guarantees case, a synthesis condition involving LMIs can be for-
mulated (see [13] for further details).
1.3 Model Recovery Anti-windup
This class of anti-windup algorithms, also referred to as L2 anti-windup, was first proposed
in [11], [50]. The motivating objective is to strive for recovering the unconstrained plant
12
1.3. Model Recovery Anti-windup
model as seen from the unconstrained controller, in order to prevent the system from
misbehaving when saturation takes place. Differently from the DLAW schemes presented
in 1.2, this strategies can be applied to nonlinear systems combined with nonlinear con-
trollers, since, as it will be clear in the following, model recovery anti-windup architecture
is completely independent of the controller dynamics. For what concerns the block dia-
gram representation of the augmented system, the general structure showed in Fig. 1.2,
specializes into that in Fig. 1.4(a), where the above mentioned strategy to achieve the
recovery objective, can be clearly noted. It consists in incorporating the plant dynamics
in the anti-windup compensator, modifying the input of the unconstrained controller from
y to y − yaw, where yaw is the output of the anti-windup compensator. At the same time
the saturation nonlinearity input is changed from the controller output yc to yc+η, where
η can be regarded as a supplemental anti-windup signal gathering the degrees of freedom
in the anti-windup system design. Since the anti-windup signals yaw, η additively affects
the unconstrained controller input and output, MRAW solutions belong to the family of
external anti-windup schemes.
Assuming perfect knowledge of the plant dynamics, the augmented system can be equiv-
alently represented as the cascade structure of Fig. 1.4(b), which underscores how the
aim of this anti-windup architecture is to keep track of what the closed loop response
would be without saturation constraints. In particular, the performance output zaw of
the anti-windup unit, can be regarded as a measure of the mismatch between the uncon-
strained variable zuc and the signal z corresponding to the augmented saturated system.
Hence, this kind of approach it’s mainly oriented to the unconstrained response recovery
objective. In this respect, all the anti-windup algorithms belonging to this framework, are
devoted to design the signal η to steer the state of the anti-windup system to zero, since
it captures the mismatch between the unconstrained and the augmented saturated system
behaviors.
In order to highlight these properties for the most general formulation of the MRAW
Plant
Unconstrained
Controller
w
sat(u) y
u
w
z
Plant Model+
-
-
+
zaw
yaw
yc
+
+
+ +
η
(a) Model recovery anti-windup scheme.
Plant
UnconstrainedController
w
yuc
u
w
zuc
Plant Model+
-
zaw
yaw
yc
+
η
+
(b) MRAW Cascade Structure.
Figure 1.4: Model recovery anti-windup block diagrams.
13
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
structure consider the, possibly nonlinear, plant
x = f(xp, u) + ν1
y = h(xp, u) + ν2(1.31)
and assume an unconstrained control law has been designed for the following model, related
to the actual plant (1.31)
x = F (xp, u) + ν1
y = H(xp, u) + ν2.(1.32)
The basic MRAW structure can be then built as follows
• Augment the system with the plant order dynamics;
xaw = f(xp, u)− F (xp − xaw, yc)
yaw = h(xp, u)−H(xp − xaw, yc)(1.33)
• change the unconstrained controller input from uc = y to uc = y − yaw;
• change the plant input from u = yc to u = yc + η.
By (1.31), (1.33), after some computations it turns out
˙︷ ︸︸ ︷
xp − xaw = F (xp − xaw, yc) + ν1
uc = y − yaw = H(xp − xaw, yc) + ν2
(1.34)
therefore the unconstrained controller is actually enforced to “see” a system evolving
according to the dynamics (1.32), for which it was designed. By virtue of this property, we
can conclude that the anti-windup unit is able to prevent the controller misbehaving when
inserted in the constrained loop. From a quantitative standpoint, it’s easy to verify that
xp−xaw is the unconstrained system state trajectory, thus, steering the anti-windup system
states to the origin, allows to completely recover the unconstrained plant behavior. Hence
the main objective in the design of the degree of freedom lying in η, is to drive xaw to zero
whenever it is possible, or at least make it minimal according to some size measurement
index. An drawback of MRAW approach is that is assumes a perfect knowledge of the
plant model, which is exploited in (1.33) to produce the cancellation needed to recover the
unconstrained system as seen by the original controller, leading to the cascade structure
of Fig. 1.4(b). Obviously this cancellation is not robust, and arbitrary small uncertainties
on the plant model impair the recovery property. For this reason MRAW can be used only
when the accuracy in the plant model is high, even though some margin of robustness
can be provided by a proper design of the function η, in order to handle the interaction
between the anti-windup compensator and the unconstrained control loop perturbation,
the tolerable mismatch, between the actual and the model plant dynamics, depends on
the specific problem.
Now consider the case of linear systems, characterized by the dynamics reported in (1.7),
i.e. with f(xp, u) = Apxp + Bpusat(u) + Bpww, h(xp, u) = Cpxp + Dpusat(u) + Dpww ,
14
1.3. Model Recovery Anti-windup
in this case the plant model (1.32) used for unconstrained control design and anti-windup
purposes becomes
xp = Apxp +Bpuu+Bpww
y = Cpxp +Dpuu+Dpww
z = Czxp +Dzuu+Dzww
(1.35)
while, according to the previously defined procedure, the corresponding anti-windup sys-
tem equations are
xaw = Apxaw +Bpu[sat(η + yc)− yc]
yaw = Cpxaw +Dpu[sat(η + yc)− yc]
zaw = Czxaw +Dzu[sat(η + yc)− yc].
(1.36)
Note that, differently for the general nonlinear case, for linear systems, no measurements
of the plant state components are needed to construct the MRAW unit. Moreover, by the
linear compensator state equation, it can be further analyzed how the signal η and the
unconstrained control input yc affect the anti-windup state xaw. In brief the anti-windup
goal can be interpreted as a bounded stabilization problem; i.e. η has to be selected in
order to drive to zero, or to keep small xaw in spite of the signal yc. In this context the
unconstrained controller output yc can be regarded as a sort of disturbance, that enters the
saturation function along with η, shifting the saturation levels and making the nonlinearity
time-varying. This control problem has been extensively considered in the literature, and
several solutions have been made available within the MRAW architecture in order to
improve the anti-windup system performance. Here just a few algorithm based on linear
compensators in the form (1.36) with η computed as a linear feedback form
η = Fxaw +G[sat(η + yc)− yc] (1.37)
and exponentially stable linear plants are presented, in order to show how also the MRAW
problem can be formulated by means of LMI constrained optimization problems, at least
in it’s simplest version. However in the literature non trivial extensions to unstable plants
([51]), possibly involving nonlinear laws for the signal η ([52]), have been proposed, along
with MRAW solution for special classes of nonlinear plants ([53]).
Constructive design algorithms are usually laid down considering the anti-windup com-
pensator (1.36) expressed in the equivalent form
xaw = Apxaw +Bpuη +Bpu(sat(u)− u)
yaw = Cpxaw +Dpuη +Dpu(sat(u)− u)
zaw = Czxaw +Dzuη +Dzu(sat(u)− u)
η = (I −G)−1Fxaw + (I −G)−1G(sat(u)− u)
(1.38)
where the interconnection law u = η+yc has been exploited to explicit the signal η. When
G 6= 0 an implicit loop need to be solved in order to implement the scheme, hence the
design algorithm have to ensure also well-posedness of this algebraic loop.
The first simple algorithm example is able to guarantee global exponential stability of
15
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
the constrained system, provided that the plant is exponentially stable, even if no other
performance indexes are optimized, it’s used in some practical application to obtain high
performance anti-windup behavior. The simples algorithm consist in setting η = 0, hence
no degree of freedom are exploited in the anti-windup design, and by(1.38), it’s straightfor-
ward to verify that the resulting compensator dynamics will be an exact copy of the plant.
Thus the so-called internal model control-based anti-windup strategy [54] is obtained. It’s
obvious that this technique relies on the plant stability properties, furthermore no per-
formance measure are optimized. More evolved technique, where the design of matrices
F , G can be cast into LMI constrained optimization problem have been proposed; among
those the following example [55] pursues the goals of global exponential stability of the
closed-loop augmented system and minimization of the following natural cost function
J =
∫ ∞
0xTawQpxaw + ηTRpηdt (1.39)
with Qp, Rp two positive definite matrices, chosen as design parameters, in a typical LQ
control design fashion. The algorithm relies on the global sector characterization for the
saturation function (see A.5), and it can be outlined as follows
• Select two positive definite matrices Qp, Rp, and solve the following eigenvalue prob-
lem (EVP);
minQ,U,X1,X2
α
s.t.
[
QATP +ApQ BpuU +XT
1
UBTpu +X1 X2 +XT
2 − 2U
]
< 0
QATP +ApQ+ 2BpuX1 Q XT
1
Q −Q−1p 0
X1 0 −R−1p
< 0
[
αI I
I Q
]
> 0, Q > 0, U > 0 diagonal
(1.40)
• Select η as in (1.37), with F = X1Q−1, G = X2U
−1, and construct the anti-windup
linear filter (1.38).
The first LMI in (1.40) ensures global quadratic stability w.r.t a Lyapunov candidate
V = xTpQ−1xp of the augmented closed-loop system, while the other two inequalities
express the minimization on the LQ index (1.39). Also in this case the plant stability is a
necessary condition for the feasibility of the above problem, however the method can be
rearranged to provide local guarantees for unstable plants (see [12] chapt. 7).
Another interesting approach was presented in [56]; it relies on the hypothesis that in
many cases the controller output in the unconstrained control loop, that by means of the
signal yaw will coincide with yc in the augmented system, converges to small values inside
the saturation linear region after a transient phase, since the unconstrained controller is
commonly designed to achieve fast convergence performance of the unconstrained loop.
16
1.4. Command Governor
Bearing in mind the previous considerations on how the unconstrained controller output
yc acts on the anti-windup dynamics (in the linear case), yc can be thought as a pulse
disturbance driving xaw away from the origin. Hence it seems reasonable to minimize
the integral of the performance output mismatch zaw = z − zuc forcing it to be smaller
than the initial condition size of the anti-windup system state xaw. Furthermore since yc
in (1.36) is multiplied by Bpu, only the initial conditions belonging to the image of Bpu.
Formally, F , G are selected in order to minimize γ in the following inequality
∫ T
0zaw(t)
T zaw(t)dt ≤ γ|xaw(T )|2 (1.41)
where T is the smallest time such that the control action returns into the saturation
linear region ∀t ≥ T , and xaw(T ) ∈ Im(Bpu). The measure above represent a sort of
H2 performance index, related to the unconstrained response recovery, moreover global
stability of the augmented system can be ensured applying the following procedure
• Solve the EVP, in the variables Q = QT > 0, U > 0 and diagonal, β > 0, X1, X2
minQ,U,X1,X2
β
s.t.
QATp +ApQ+ 2BpuX1 XT
1 −BpuX2 −BpuU XT1
−UBTpu −XT
2 BTpu +X1 −2U − 2X2 XT
2
X1 X2 −I
< 0
[
QATp +ApQ+ 2BpuX1 QCT
z +XT1 D
Tzu
CzQ+DzuX1 −I
]
< 0
[
βI BTpu
Bpu Q
]
> 0;
(1.42)
• Select η as in (1.37), where F = (I +X2U−1)−1X1Q
−1, G = (I +X2U−1)−1X2U
−1,
and consturct the anti-windup compensator as (1.38).
Also in this case, the first LMI condition, provides global quadratic stability w.r.t the
Lyapunov candidate V (xp) = xTpQ−1xp, while the other two conditions are related to
the performance index optimization. Furthermore the LMIs ensures that matrix (I +
X2U−1) is non singular, hence the algorithm can be completed with the anti-windup gain
computation.
1.4 Command Governor
In this section the so called command or reference governor (CG) is discussed; differ-
ntly from the two previously presented methodologies, this strategy does not fall into
the general paradigm reported in Fig. 1.2. The typical command governor control block
diagram is reported in 1.5; in plain words, the reference feeding the unconstrained con-
troller, designed disregarding constraints, is, if necessary, a suitable elaboration, performed
by a device added to the forward path, of the original reference, with the specific goal to
17
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
PlantUnconstrainedController
gc
u
y
CommandGovernor
r
d
xpx cxcl
Figure 1.5: Command governor scheme.
prevent constraints violations, and possibly keep good tracking performance. Other differ-
ences with respect to DLAW and MRAW approaches are that, in addition to control inputs
limitations, command governor paradigm can explicitly handle also state constraints. Fur-
thermore, the anti-windup unit design is usually performed in the discrete time domain
exploiting a receding horizon approach typical of the MPC framework. Historically this
technique was proposed in the 90’s, first for unperturbed, linear time invariant systems
[57], [58], then extensions to bounded unknown disturbances [8], and generic linear time-
varying systems [7], [59] were presented, finally also solutions for nonlinear constrained
systems have been considered [60].
Here the basics of this solution are briefly outlined considering the simple case of LTI
systems in presence of unknown bounded disturbances. The following closed-loop system,
regulated by an unspecified suitable unconstrained controller, is assumed
xcl(t+ 1) = Aclxcl(t) +Bgg(t) +Bdd(t)
z(t) = Cxcl(t)
c(t) = Hxcl(t) + Lg(t) + Ld(t)d(t)
(1.43)
where xcl = [xp xc]T ∈ R
n is the state including the plant and controller states, g(t) ∈ Rm
is the command governor output, i.e. a properly modified version of the original reference
r(t). While d(t) is an external disturbance which is supposed to lie in a known convex
compact set D containing the origin in its interior, z(t) ∈ Rm is the performance output,
which is required to track r(t), and c(t)the vector lumping the system constraints viz.
c(t) ∈ C, ∀t > 0 (1.44)
where C a defined compact convex set. Then, under the following common hypothesis
1. System (1.43) is asymptotically stable;
2. System (1.43) has a unitary DC-gain, i.e. C(I −Acl)−1Bg = Im.
the command governor anti-windup problem can be formulated as: design a memoryless
function of the current state and reference
g(t) = g(x(t), r(t)) (1.45)
18
1.4. Command Governor
so that g(t) is the best pointwise approximation of r(t), under the constraints (1.44), and
for any d(t) ∈ D. Actually, the following additional objectives are commonly pursued
1. g(t) → r, if r(t) → r ≡ const., with r the best feasible approximation of r;
2. g(t) = r after a finite time if r(t) ≡ r.
The requirements above can be interpreted in the light of the objectives stated in 1.1; the
small signal preservation and the unconstrained system response property are somewhat
included in the requirement to find the best feasible approximation of the original reference.
Whenever r(t) does not violate any constraint, the command governor is requested to be
silent, i.e. g(t) = r(t). While when constraints are active, the reference, and then the
system response z(t), are the best possible approximations of the signals produced by an
ideally unconstrained system. As regards the stability objective, it will be elaborated in
the following.
The command governor unit design is commonly carried out consider a class of constant
command sequences v(·, ν) = ν, the idea is to compute at time t, given the current state
and reference, a constant sequence g(t+ k|t) = ν that, if used as the system set point for
the subsequent instants t+k, would avoid constraints violations, furthermore the distance
of the system evolution from a constant reference value r(t) would be minimal. This
procedure can be clearly cast in the receding horizon philosophy, by applying just the first
term of the sequence g(t) at time t, and then recompute the sequence at t+1, given x(t+1),
and r(t+1). It’s further to remark that, differently from a general model predictive control
solution, here the stabilization of the system is left to a pre-designed controller, while the
duty of the CG unit is to handle the system constraints, this leads in general to simpler
constrained optimization problems, at the cost of some performances. However, the CG
anti-windup paradigm seems more suitable then the MPC when saturation is not expected
to occur frequently during the system nominal operation, or a controller has already been
designed.
Now we briefly define a common procedure ([59]) to design a command governor for
system (1.43); in order to deal with disturbances, system linearity is exploited to separate
the effects of d(t) from those of the initial conditions and inputs, e.g xcl = xcl+ xcl, where
xcl is the disturbance free component depending only on xcl(0) and g(·), while xcl is the
state response due to d(t) with xcl(0) = 0, g(t) ≡ 0. Same reasoning can be applied to
c(t) = c(t)+ c(t). For design purpose, the disturbance free steady-state solutions of (1.43)
correponding to g(t) ≡ ν is denoted as
xclν := (I −Acl)−1Bgν
zν := C(I −Acl)−1Bgν
cν := H(I −Acl)−1Bgν + Lν
(1.46)
19
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
and the following sets are defined
C0 := C ∼ LdDCk := Ck−1 ∼ HAk−1
cl BdD
C∞ :=∞⋂
k=0
Ck(1.47)
with A ∼ B := a ∈ A : a+ b ∈ A, ∀b ∈ B. It can be verified that the sets Ck represent
non-conservative restrictions of C such that if c ∈ C∞, then c(t) ∈ C. Thus the design can
be carried out considering the disturbance free evolutions and a worst case scenario, by
taking the sets
Cδ := C∞ ∼ Bδ, Wδ :=
ν : cν ∈ Cδ
(1.48)
where Bδ := c : ‖c‖ ≤ δ. Roughly speaking, Wδ contains the commands ν producing a
steady-state which fulfills the constraints with margin δ. Now define the set of all constant
commands belonging to Wδ, that satisfies the constraints also during transient, as
V(xcl) :=
ν ∈ Wδ : cν(k, ν, xcl) ∈ Ck, ∀k > 0
(1.49)
where
cν(k, xcl, ν) := H
(
Akcl +
k−1∑
i=0
Ak−i−1cl Bgν
)
+ Lν (1.50)
is the disturbance-free trajectory of c, at time k, under a constant command ν. There-
fore the following inclusion holds: V(xcl) ⊂ Wδ. Finally, the robust command governor
problem can be formulated as
g(t) = argminν
(ν(t)− r(t))TQ(ν(t)− r(t))
s.t. ν ∈ V(xcl)(1.51)
where Q = QT > 0 is a design parameter penalizing the reference components.
If the assumptions 1 − 2 are satisfied by system (1.43), and Vxclis non empty, then the
following properties hold for the considered CG selection rule [7]:
• The minimization problem (1.51) is feasible and convex i.e.: there exists a unique
optimal point and, if V(xcl(0)) is non-empty, then V(xcl(t)) in non empty along the
trajectories generated by the solutions of (1.51);
• There exists a finite constraints horizon k such that, if c(xcl, k, ν) ∈ C ∀ k = 1, . . . , k,
then c(xcl, k, ν) ∈ C ∀ k > k. That is, the set V(xcl) is finitely determined;
• The overall system is asymptotically stable and, if r(t) ≡ r, g(t) approaches r, or its
best feasible approximation r, in finite time.
Here the complete proof of the above properties, along with the procedure to compute
the value of k to be adopted in the receding horizon algorithm for 1.51, is omitted (see
[7], [59] for details). However, the stability property of the overall constrained system
20
1.4. Command Governor
deserves attention, so that the analysis of the CG approach in terms of the three qualitative
objectives stated in 1.1 is complete. Note that, since the modified reference g(t) depends
on the current state xcl(t), an extra feedback loop is somewhat introduced, hence stability
does not trivially follows by the unconstrained controller properties. Stability of the CG
method is easily verified if a constant reference r(t) ≡ r is considered. Taking as candidate
Lyapunov function the value function
V (t) = (ν(t)− r)TQ(ν(t)− r) (1.52)
if xcl(t + 1) satisfies the state equation in (1.43), it can be proved that V (t) ≥ V (t + 1)
∀ t > 0. This claim can be motivated ad follows; since ν(t) is not in genral an optimal
point for (1.51) at time t+ 1, there exists a feasible ν(t+ 1) such that
(ν(t+ 1)− r(t+ 1))TQ(ν(t+ 1)− r) ≤ (ν(t)− r(t))TQ(ν(t)− r) (1.53)
thus, as V (t) is non negative and non increasing it has a finite limit
V (∞) = limt→∞
(ν(t)− r)TQ(ν(t)− r) (1.54)
which proves stability of the overall closed-loop system.
Beside stability and the previously defined features, CG approach is also endowed with the
so called viability property [7]: given an admissible initial condition xcl(0), there exists a
finite number of constant commands v(·, νi) = νi which, concatenated with finite switching
times, can steer the system state from xcl(0) to any xclν+∆∞, with ν ∈ Wδ. Where ∆∞ is
the Hausdoff limit [8] of the following sequence ∆0 = 0, ∆k = ∆k−1+A−1cl BdD. By stabil-
ity properties of Acl, the limit exist and it coincides with the smallest closed set containing
the state evolution xcl(k), forced by all the possible sequences diki=0 ⊂ D. Roughly
speaking, viability property ensures that, starting from any feasible initial condition, any
admissible disturbance-free steady state condition xclν , ν ∈ Wδ can be approached at a
possibly small distance, in finite time and without violating constraints.
The command governor approach seems to merge the benefits of the DLAW and MRAW
paradigms, since it can explicitly account for partially known disturbances or model uncer-
tainties, it’s independent from the pre-designed unconstrained controller, and it can apply
to nonlinear plants. However its main drawback lies in fact that the reference modifica-
tion has to be computed by solving an on-line constrained optimization problem which,
for nonlinear plants, or nonlinear constraint functions, is in general not convex. Hence
the computational burden required to find the global optimal solution can significantly
increase w.r.t the linear case. Moreover, also a procedure to determine the control horizon
k is needed, in [57] a suitable algorithm is proposed, relying on the solution of a sequence
of mathematical problems, which are convex only in the case of constraints function affine
in the state variables. Even if this algorithm can be run off-line, and a finite k is ensured
to exist also in the nonlinear case [60], determining such k can be a difficult task also for
rather simple systems. Finally the value of k is problem-specific and it obviously affects
the optimization problem dimensions, hence even in the simple cases of linear plants with
21
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
affine constraints, for which (1.51) reduces to a quadratic programming problem, complex-
ity can stem from a large horizon on which to evaluate the constraints at each step. Due
to this reasons, when only input constraints are concerned, DLAW and MRAW strategies
can lead to simpler high performance anti-windup solutions.
1.5 Novel Anti-Windup strategy: general idea
Here an original anti-windup strategy, first sketched in [30], is presented in it’s main fea-
tures, considering a rather general class of, possibly nonlinear, systems. The obtained
results will be exploited in chapter 2, where a saturated control solution for a class of
power converters used as active filters is presented. In particular, the guidelines provided
in this section will be adopted, and further developed, to design the anti-windup unit for
the current controller of the above mentioned class of power electronics systems.
As usual in the anti-windup framework, we assume that a proper controller has been
designed in order to ensure the desired stability and tracking properties for a virtual
unconstrained system. The key objective of the proposed approach is to construct an
anti-windup system which is able to preserve, whatever the original controller is and un-
der any condition, the closed-loop unconstrained system tracking error dynamics. In this
way all the results provided by the unconstrained control law design, would still be valid
when saturation is concerned.
The proposed way to achieve this task, is to modify the reference to be tracked by the
closed-loop system, and combine the reference modification with a suitable feed-forward
action, related to it, such that input saturation is prevented. In this way, it looks rea-
sonable and feasible to impose no modification on the tracking error dynamics, while the
control input windup effect is avoided.
This philosophy shows some similarities to the command governor framework, however
the crucial difference lies in the main objective of the scheme; here the priority is given
to the closed-loop tracking behavior preservation, at the cost of some optimality in the
obtained reference correction. In this respect, the reference correction is not produced by
a memoryless device, but by a suitable “additional dynamics” injected into the closed-
loop system, in order to generate a smooth and feasible modified reference, according to
the system relative degree. Furthermore, no on-line solution of constrained optimization
problems is required. Hence, the obtained reference modification will not in general be
minimal in the sense defined in 1.4, on the other hand, this strategy can lead to simpler
effective solutions for nonlinear systems, and or when the specifications cannot be trivially
cast into the CG standard framework.
For the sake of completeness, it’s worth to cite that also in the so-called conditioning tech-
nique, proposed by Hanus et. al in [4], [5], the input constraints are handled by producing
realizable reference signals, however the main purpose of that approach was to reduce the
effect of saturation on the controller behavior, preserving the controller states coherence.
22
1.5. Novel Anti-Windup strategy: general idea
Now consider the following simple but rather significative class of systems
xp = f(xp) + u(t)
y(t) = xp(t)(1.55)
where xp(t), u(t), y(t) are the plant state, control input, and controlled output respec-
tively, and the following properties hold
• f(·) is a smooth vector fields;
• system (1.55) is square, i.e. it has the same number of inputs and controlled outputs,
xp(t), u(t) ∈ Rn, and functionally controllable according to the definition given in
[61];
• the control input vector u(t) is constrained to lie in a connected compact set U ⊂ Rn.
It’s further to notice that (1.55) can be seen as a particular case of the so-called generalized
normal form systems [62],[63], for which functional controllability, along with the relative
degree, that in this simple case is just one, are well defined and can be checked by means
of formal procedures (see [63] and reference therein).
Bearing in mind this considerations, define a sufficiently smooth reference x∗, more pre-
cisely belonging to the class of C1 functions, bounded with bounded first derivative, and
the corresponding tracking error variables x = xp−x∗. According to the functional control-lability hypothesis, there exist a control input and a proper set of initial conditions, such
that the reference x∗ can be exactly reproduced at the output. Now assume this steady-
state control action can be robustly recovered, with the desired convergence properties,
by the following unconstrained feedback law
u = c(x∗, x, xc)
xc = m(x∗, x, xc)(1.56)
namely, the resulting closed-loop tracking error dynamics
˙x = f(x∗ + x) + c(x∗, x, xc)− x∗
xc = m(x∗, x, xc).(1.57)
is stabilized at the origin with the required properties.
Now the limitations on the control input authority need to be addressed. As claimed
before, the basic idea is to add a reference modification xaw , and a suitable feed-forward
action gaw(xaw, ·), capable of achieving the following objectives
• u(t) ∈ U , ∀t;
• for each condition s.t. c(·) ∈ U , xaw is null or it tends to zero in some way;
• the new error ˜x = xp−x∗−xaw dynamics are exactly the same as those reported in
(1.57), replacing x with ˜x.
23
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
The first item of the list is similar to the main CG objective, that, differently from DLAW
and MRAW schemes, aims to prevent saturation rather then handling its effects on the
unconstrained controller, while the second goal can be clearly related to the small signal
preservation and unconstrained response recovery properties defined in 1.1. As regards the
overall system stability, it’s easy to guess that it will be structurally ensured if the third
objective, which is the peculiar feature of the proposed approach, is fulfilled. Indeed, if the
unconstrained tracking dynamics form is preserved, also its stability properties will still
hold despite the anti-windup augmentation. It’s worth to remark that stability turns out as
a structural consequence of this predefined objective, which is in general more demanding,
and focused to guarantee high performance anti-windup behavior to the system.
Starting from this last point, define the modified tracking error dynamics ˜x by
˙x = f(˜x+ x∗ + xaw) + c(x∗, ˜x, xc) + gaw(xaw, ·)− x∗ − xaw
xc = m(x∗, ˜x, xc)(1.58)
where the original controller has been modified as uaw = u + gaw, and, in u defined by
(1.56), x has been replaced by ˜x, but x∗ has not be replaced by x∗+xaw. Hence, assuming
a perfect knowledge of the plant dynamics, by simple computations, it can be verified that
defining gaw(xaw, ·) as
gaw(xaw, ·) = f(˜x+ x∗)− f(˜x+ x∗ + xaw) + xaw (1.59)
dynamics (1.58) are made identical to (1.57). It’s further to remark that here the functional
controllability hypothesis is exploited, since gaw(xaw, ·) in (1.59) is based on the system
right inverse, and it embeds the control action needed to perfectly track any differentiable
reference modification xaw.
Thus the next step is to design such an xaw in order to fulfill the remaining of the above
mentioned objectives i.e.
a) uaw = c(x∗, ˜x, xc) + g(xaw, ·) ∈ U ;
b) xaw is bounded and moves towards zero, while c(x∗, ˜x, xc) ∈ U .
The actual xaw design is related to the specific system, however, by (1.59) it can be noted
that, in general, xaw can be considered as the actual anti-windup steering input, hence
an additional reference dynamics needs to be managed. This feature can be equivalently
motivated recalling the basic objective of preserving the closed-loop system tracking be-
havior, from which equation(1.59) stems from; for this purpose a r-times differentiable
reference modification, where r is the system I/O relative degree, needs to be generated
in order to be perfectly tracked by the closed-loop systemd.
In this respect, since the considered system relative degree is one, the anti-windup unit
has to be completed by designing the following dynamics:
xaw = h(U , c(·), x∗, xp, xaw) (1.60)
24
1.5. Novel Anti-Windup strategy: general idea
which is required to guarantee objectives a) and b). Note that (1.60) is a sort of internal
dynamics with respect to the output tracking error ˜x, that needs to be properly stabilized
in order to ensure objective b). Therefore, even if it seems reasonable to fulfill objective
a) trying at the same time to minimize g(xaw, ·), i.e. this term should be null, whenever
c(·) ∈ U , while it should keep the overall control action on suitable points of the feasibility
set boundary ∂U , whenever c(·) /∈ U , this choice could in principle lead to to poorly
damped or even unstable anti-windup dynamics (1.60). Hence a degree-of-freedom should
be preserved, in order to and ensure a bounded and “reasonable” behavior of xaw according
to objective b). In this respect, a possible solution to cope with such issue, is to formulate
a sort of constrained minimum-effort control problem. Alternatively, a simpler, even if
“suboptimal”, strategy is to save a part (possibly small) of the control action to shape the
xaw dynamics. This second procedure is here outlined as follows: given raw > 0, define
Ur = u ∈ U s.t. dist(∂U , u) ≥ raw . (1.61)
where dist(x, S) denotes the distance of x from the set S defined, according to [11], as
dist(x, S) = miny∈S‖x− y‖. Note that Ur lies in the interior of U and, between ∂Ur and
∂U there is a “stripe” of width raw. Obviously, raw cannot be too large, otherwise U = ∅;but, as it will be clear in 2, “small” raw will be considered for practical application of the
proposed procedure.
The next step is to redefine xaw in (1.60) as the sum of two terms
xaw = h1(Ur, c(·), x∗, x, xaw) + h2(·) (1.62)
then, substituting in (1.59) yields
gaw(xaw, ·) =g1(·)
︷ ︸︸ ︷
f(˜x+ x∗)− f(˜x+ x∗ + xaw) + h1(·)+h2(·) (1.63)
u(t) = c(·) + g1(·) + h2(·). (1.64)
Then, h1 can be simply designed in order to minimize ‖g1‖ giving (c(·) + g1(·)) ∈ ∂Ur and
neglecting any issue related to xaw behavior; while h2 has the task to properly shape xaw
dynamics. In order to satisfy objective a), i.e. to guarantee u(t) ∈ U , it’s easy to verify
that h2 need to be saturated as: ‖h2‖ ≤ raw. Obviously, raw should be selected as small
as possible in order to make Ur very close to the original U and not to “waste” too much
control authority with this a-priori preservation.
With all these results at hand, it’s worth to underscore that the anti-windup design is
completely independent form the adopted unconstrained control solution; the only natural
requirement is the stability of the saturation-free closed-loop tracking dynamics. Further-
more, the resulting error dynamics ˜x, and the additional anti-windup dynamics xaw are
structurally decoupled, whatever the anti-windup dynamics are. As a consequence, the
additional dynamics can be stabilized and designed to meet the above mentioned specifi-
cations, without affecting the closed-loop error dynamics. On the other hand this strong
decoupling property, along with the unconstrained error dynamics form recovery, holds
25
Chapter 1. MODERN ANTI-WINDUP STRATEGIES
only if perfect knowledge of the plant model is available. If it is not the case, and a nomi-
nal model f is used to design the unconstrained controller and to compute gaw(xaw, ·), it’sstraightforward to verify that the ˜x dynamics, instead of being identical to (1.57), would
be perturbed by the plant model mismatch f(˜x+ x∗ + xaw)− f(˜x+ x∗ + xaw), since the
cancellation provided by the term gaw would not be perfect. Hence decoupling and the
original tracking behavior recovery properties, would be destroyed by model uncertainties.
This drawback is similar to what discussed in 1.3 for the MRAW framework, hence similar
considerations apply. Obviously also this scheme is not tailored to provide robustness,
but it’s mainly focused on high performance anti-windup, and should be adopted only
when an accurate system model is available. On the other hand, for linear plants and
parametric uncertainties some robustness properties can be enforced by relying on input-
to-state stability (with restriction on the initial state and the input amplitude) of either
the unconstrained closed-loop dynamics (w.r.t f(˜x+x∗+xaw)− f(˜x+x∗+xaw)) and the
additional anti-windup system (w.r.t a sort of mismatch signal between the unconstrained
control action and the saturated one).
26
Chapter 2
Saturated Nonlinear Current
Control of Shunt Active Filters
In this chapter the control of a particular kind of Active Power Filter (APF),
so-called Shunt Active Filters (SAF), aimed to compensate for harmonic cur-
rent distortion, is addressed, formally accounting for input and state constraints.
The system model and the overall control objectives are introduced, along with a
possible robust nonlinear unconstrained control solution. Then the filter control
input and current constraints are formally dealt with; the proposed anti-windup
strategy is specialized for the system current control, and suitably combined with
a current saturation strategy.
2.1 Introduction
Electric pollution in the AC mains is a common and significant issue in industrial plants,
since it worsen the system power factor, and gives rise to additional power losses, nega-
tively affecting the plant electrical energy exploitation, and increasing the operating costs.
Beside these economic motivations, a high level of electric pollution, could lead to sys-
tem malfunctioning or even damages to other plant equipments that are connected to the
same portion of the perturbed line grid. Electric pollution is mainly caused by reactive
and harmonic distortion currents injected into the mains by the input stage of industrial
nonlinear loads, e.g. rectifiers or motor drives.
Traditionally, passive filtering components have been adopted to compensate for harmonic
distortion, however they are affected by several drawbacks: high sensitivity to network
impedance variation and environmental conditions, a priori defined filtering frequencies
with poor tuning flexibility, possible safety critical behavior due to resonance phenomena.
In order to overcome these limitations, in the last decades, driven also by the fast growth
in power electronics and processing units technology, a remarkable research attempt has
been devoted to the study of the so-called Active Power Filters (APFs), both from a
theoretical and technological point of view (see [64], [65], [66] for a quite comprehensive
27
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
vma
L1 L2 L3
LoadN
C
v
ima
ia
K
ua
imb
imc
vmb
vmc
ib ic
ila
ilb
ilc
SAF ub
uc
Figure 2.1: Shunt Active Filter topology.
overview). These devices are able to properly work in a wide range of operating condi-
tions, providing better performance and overtaking intrinsic limitations of passive devices;
they are far less sensitive to network impedance variations and they can be tuned onto
different frequencies just varying some software parameters. Furthermore, also reliability
is improved, since resonance phenomena are avoided and an active diagnosis system can
be implemented on the control processor to monitor the system variables and adopt some
recovery strategy in case of faulty conditions.
2.1.1 SAF saturated control strategy motivation
In this chapter a particular class of APFs, the so-called Shunt Active Filters (SAF) is con-
sidered. Roughly speaking, the main purpose of this kind of power converters is to inject
proper currents into the mains, in order to cancel, partially or totally, the effects of the
nonlinear load current harmonics. The considered filter topology is reported in Fig. 2.1,
it is based on a three-phase three-wire AC/DC boost converter [67] connected in parallel
between the mains and the nonlinear load. The main energy storage element is a DC-bus
capacitor, while the inductances are exploited to steer the filter currents by means of the
converter voltages. The switching devices of the three-legs bridge (also called “inverter”)
are usually realized by IGBTs and free-wheeling diodes. From a control theory standpoint,
the main distinguishing mark, characterizing the system compensation performance, is the
filter currents control algorithm. In this respect, several approaches have been presented
in literature; in [68] the high performances of hysteresis current control are exploited, in
[69] current predictive control has been proposed, while in [70], [31] the internal model
principle has been exploited to design a robust current control solution.
Usually, for this kind of systems, the controller design is carried out disregarding con-
straints that, inthis case, concern the maximum voltage on the inverter legs and the
maximum current that the converter switching devices can drain. These limitations can
be partially handled by means of a suitable device sizing procedure ([71], [32]), that en-
sures to avoid constraints violation, provided that the system operates in a predefined
28
2.1. Introduction
set of nominal conditions. Anyway, this is not sufficient to achieve high performance and
reliability indexes for actual industrial applications, since in real plants there are several
operating conditions (such as line voltage amplitude and load current profile variations)
requiring to exit from the nominal working area during transients, or even permanently.
Common devices just turn off when facing such conditions, impairing their availability.
While an effective control under saturation conditions would guarantee to withstand off-
nominal transients and extend the system operating range, ensuring, at the same time,
stability and a graceful performance degradation. This would enable also to reduce over-
sizing, to meet modern regulations requirements and to improve reliability and availability
of such system, particularly in complex, and possibly “smart”, grid topologies, where non
standard operating scenarios occurs more frequently then in traditional power networks.
Since several high performance unconstrained control solutions are available, and satu-
ration is not expected to be a common situation for the nominal system conditions, the
anti-windup framework is the most suitable to cope with saturated control of this class of
power filters. To this aim, here the internal model based solution proposed in [31], will
be adopted as a benchmark unconstrained control law, then the strategy defined in 1.5
will be specified, extending the design method presented in [30], in order to achieve the
following qualitative objectives
a) Maximum enlargement of the system working region;
b) Preservation of the original unconstrained current and DC-bus voltage dynamics
under saturation;
c) Avoidance of additional harmonics injection by the anti-windup unit;
d) Minimization of the reference modification needed to comply with the previous re-
quirements.
It’s worth noting that objective d) is different from the common requirement of standard
CG approaches; here minimizing the reference modification is not the maximum priority
goal, but it is subject to the previous requirements, especially b) and c). In other words,
provided that no spurious harmonics are injected into the system and the unconstrained
system tracking properties are maintained, the resulting reference modification can be not
minimal in the sense discussed in 1.4.
However the system is also subject to a maximum current constraint, which clearly affects
the anti-windup unit, since it is based on a proper current reference modification. As
it will be clear in the following, the maximum current limit can be regarded as a state
constraint. Hence, Command Governor or a Model Predictive Control methodologies may
seem more suitable to face both input and state constraints by means of a one step proce-
dure. Nevertheless, the above outlined objectives are rather unusual for receding horizon
base frameworks. Beside the plant nonlinearity, that would demand a significant compu-
tational effort to solve on-line the constrained optimization problems arising in CG and
MPC approaches, a crucial issue in SAF control, is the stabilization of the DC-bus voltage
29
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
dynamics, which can be regarded as a non minimum phase internal dynamics with respect
to the filter currents outputs. Moreover, requirement c on the avoidance of additional
harmonic pollution injected by the anti-windup unit, is not trivial to be incorporated into
a standard MPC objective function. It is further to remark that this specification is of
utmost importance, as the main objective of SAF is to compensate for distortion, thus
new spurious currents given by anti-windup appliance are unacceptable.
For this reasons, here a multiple step approach is considered to deal with this non standard
and heavily interlaced constrained control problem; first, starting from a suitable relax-
ation, the proposed anti-windup system is added to the current controller, then, relying on
it, a suitable current reference saturation law is defined to face also the current limitations.
In this respect, the method that will be carried out along the chapter is outlined in the
following:
1. Focusing on the input saturation, and disregarding either the DC-bus dynamics
preservation under saturation, and the filter current bounds, an anti-windup unit
based on the guidelines presented in 1.5 is designed;
2. The straightforward application of the AW general method in 1.5 to SAFs is slightly
modified to take into account the DC-bus voltage dynamics preservation require-
ment.
3. Objective c is accomplished by steering the anti-windup dynamics towards a piece-
wise constant steady state current modification value. This behavior is enforced
modifying the anti-windup dynamics design, based on the periodicity of the load
current harmonics to be compensated for;
4. The effects of the previous modifications to the anti-windup units in shrinking the
maximum working region are carefully analyzed. Then, taking into account this
limitations and the maximum current limits, a current reference saturation strategy,
consisting in a suitable scaling of the current harmonics to be compensated for,
is defined in order to fulfill both the anti-windup unit objectives and the current
constraints.
Bearing in mind this outline, the chapter is organized as follows. In Section 2.2, the SAF
mathematical model is derived and the control objectives are formally stated, in Section
2.3 the features of the internal model based current control solution, presented in [70], [31],
are recalled. In Section 2.4 the system control input and current saturation constraints
are analyzed, their causes are discussed and the effects produced by practical common
saturation scenarios are shown via simulation tests. In Section 2.5, following the strategy
presente in [30], a specific anti-windup scheme is constructed to handle SAF input vector
saturation, and the improvements defined in 2−3 are introduced. In Section 2.6 item 4 of
the previous list is carried out. The control input feasibility set is analyzed, assuming the
system has been augmented with the propose anti-windup unit and and accounting for
the current constraints, then a numerically tractable optimization problem is formulated,
30
2.2. System model and control objectives
in order to compute the maximum feasible reference current. Finally, on the basis of the
maximum available reference a proper current saturation strategy. In section 2.8.2 the
proposed strategy is tested by extensive simulations of a realistic system. Section 2.9 ends
the chapter with a possible variation on the anti-windup dynamics design, which enriches
the class of generated reference modification signals.
2.2 System model and control objectives
The notation reported in Fig. 2.1 is adopted to denote the model variables; vm =
(vma, vmb, vmc)T is the mains voltage tern, im=(ima, imb, imc)
T are the mains currents,
il=(ila, ilb, ilc)T is the nonlinear load current vector, i=(ia, ib, ic)
T are the filter currents,
while v is the capacitor voltage. The inverter switches commands, that are the actual
system control knobs, are denoted with the vector u1 = (ux, uy, uz), while L and C are
the inductances and capacitor values respectively.
2.2.1 State space model derivation
Considering the inductors dynamics, the filter model can be expressed as
vma(t)
vmb(t)
vmc(t)
− L
d
dt
ia(t)
ib(t)
ic(t)
−R
ia(t)
ib(t)
ic(t)
=
ux(t)
uy(t)
uz(t)
v(t)− vNK
1
1
1
(2.1)
where R is the parasitic resistance related to the inductance L and to the cables, while
vNK is the voltage between the nodes N and K reported in Fig. 2.1. Since a PWM
(Pulse Width Modulation) strategy is usually exploited to control the inverter (achieving
the desired voltage on the converter legs as mean value in a PWM period) the above-
mentioned control inputs are constrained to lie in the range u1i ∈ [0, 1], i = x, y, z.
According to the three-wire topology, for any generic voltage/current vector x it holds∑
i=a,b,c xi = 0. Thus, summing the scalar equations in (2.1) it follows that
vNK =ux(t) + uy(t) + uz(t)
3v(t) (2.2)
now, defining
uabc = [ua(t), ub(t), uc(t)]T =
ux(t)
uy(t)
uz(t)
− ux(t) + uy(t) + uz(t)
3
1
1
1
(2.3)
by direct computation it follows [1 1 1]uabc(t) = 0 ∀t ≥ 0. For what concerns the capacitor
voltage dynamics, it can be derived considering an ideal inverter, and applying a power
balance condition between the input and the output stages of the filter. Replacing (2.2)
into (2.1), the complete filter model is
di
dt= −R
LI3i(t)−
v(t)
Luabc(t) +
1
Lvmabc
dv
dt=
1
CuTabc(t)i(t).
(2.4)
31
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
The system model can be reduced to the standard two-phase stationary planar represen-
tation of a three-phase balanced systems [72], which is obtained applying the following
coordinates transformation
iαβ(t) = [iα iβ]T =αβ Tabci(t)
uαβ(t) = [uα uβ ]T =αβ Tabcuabc(t)
vmαβ = [vmαvmβ ]T =αβ Tabcvm
αβTabc =2
3
[
1 −12 −1
2
0√32 −
√32
]
(2.5)
then, the resulting SAF dynamics are
diαβdt
= −RLI2iαβ(t)−
v(t)
Luαβ(t) +
1
Lvmαβ
dv
dt=
3
2CuTαβ(t)iαβ(t).
(2.6)
In order to simplify the control objectives definition and the controller design, it is conve-
nient to adopt a further transformation, from the two-phase current variables [iα iβ]T to
a so-called d-q synchronous rotating reference frame, aligned to the mains voltage vector,
according to the following change of coordinates
idq = [id iq]T =dq Tαβiαβiαβ
dqTαβ =
[
cos(ωmt) sin(ωmt)
−sin(ωmt) cos(ωmt)
]
.(2.7)
The previous transformation yields the following state-space model that will be considered
throughout the chapter for control and anti-windup purposes
d
dtidq =M(R,L)idq(t)−
v(t)
Lu(t) + d0
v = ǫuT idq
(2.8)
with
d0 =
[Vm
L
0
]
, M(R,L) =
[
−R/L ωm
−ωm −R/L
]
, ǫ =3
2C(2.9)
where Vm, ωm are the mains voltage amplitude and angular frequency respectively, and
u = [ud uq]T is the transformed control input vector. The same synchronous coordinate
setting can be adopted to represent the load currents; in particular, any nonlinear load
current profile can in principle be approximated as a finite sum of harmonics [73], obtaining
the following general expression
ilj = Ilj0 +
N+1∑
n=1
Iljncos(nωmt+ ψjn), j = d, q. (2.10)
It’s further to remark that, in order to represent the system variables in such synchronous
coordinates, an accurate, and possibly robust, estimation of the line phase-angle and
frequency need to be performed (see 2.7), this issue will be detailed in ch. 7, where the
nonlinear adaptive observers framework is exploited to design a robust estimation scheme
of three-phase line voltage parameters.
32
2.2. System model and control objectives
2.2.2 Control objectives
If the previously defined synchronous coordinates representation is adopted, each current
vector (i.e.. SAF current, load current, main current) is structurally split into a real, or
active, part (the d-component), and virtual, or reactive, part (the q-component) ([73]),
this simplifies the control problem definition and the controller design.
Roughly speaking, the main control objective is to steer the filter current vector idq, in
order to cancel out the undesired load harmonics at the line side. It turns out that the only
desired load component in (2.10) is Ild0, since, in the d−q reference frame it represents the
first-order harmonic aligned with the mains voltages (i.e. the mean power component),
while the remaining part of the real component ild − Ild0 and all the reactive component
ilq, are undesired terms which do not contribute to the power flow but worsen the system
power factor (see [67]). In addition, as it will detailed in chapter 3, the DC-link voltage
differential equation represents an internal dynamics with respect to idq. Therefore it
needs to be carefully managed, since, the energy stored in the DC-bus provides the control
authority to steer the filter current. Bearing in mind these considerations, the SAF control
problem can be formulated as follows.
O1) The instantaneous currents drained by the filter have to asymptotically compensate
for the oscillating component of the load real current, and for the load reactive current;
this imply that the idq subsystem has to track the following reference
i∗dq =
[
Ild0 − ild + η
−ilq
]
(2.11)
where η is an additional active current term necessary to compensate for the filter power
losses and stabilize the DC-bus voltage dynamics (see O2).
O2) Given a safe voltage range [vm, vM ], assuming v(t0) ∈ [vm, vM ], the following condition
must be fulfilled
v(t) ∈ [vm, vM ], ∀t > t0. (2.12)
The lower bound vm > 0 is chosen to satisfy a controllability constraint under nominal
working conditions, while the upper bound vM is selected according to the capacitor
maximum voltage ratings [71].
The two control objectives O1 and O2 are interlaced, and possibly in contrast each other.
However, in ch. 3 it will be showed how a suitable capacitor sizing, combined with a
voltage controller producing the additional reference current η, allows to comply with
both the objectives. In particular, the above mentioned sizing procedure can be exploited
to enforce a two time-scale separation property between the SAF currents and DC-bus
voltage dynamics. Therefore, according to Singular Perturbation Theory results (see [15]
ch. 11), the main tracking objective O1, and the voltage stabilization requirement O2,
can be separately handled by two different controllers combined in the overall structure
of Fig. 2.2. In the following the focus is put on the current control unit and its saturation
issues, while the the DC-bus voltage stabilizer is left unspecified, and it will be deeply
discussed in ch. 3. However, as reported in the outline of the proposed saturated control
33
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
SAFCurrent
Controller
+
-Voltage
Controller
+
+
-
+
v
V ∗
idq
i∗dq i∗dq
η
u
Figure 2.2: SAF Control structure.
design procedure (item 2 in 2.1.1), some adjustments to the anti-windup structure are
specifically devoted to not impair the voltage dynamics by means of the current reference
modification.
2.3 Internal model-based current controller
In this section, the robust current tracking solution, based on the internal model principle,
presented in [70] and improved in [31], is recalled, as it will be used to test the anti-
windup and current saturation strategies. For this purpose, we assume the state vector
components idq, v are available from measurement (which is usually the case for practical
filter realizations), and a suitable voltage stabilization algorithm, as those presented in ch.
3, is able to enforce v(t) ∈ [vm, vM ].
Defining the current error variables, w.r.t the reference defined in (2.11), idq=idq− i∗dq, thethe current subsystem in (2.8) can be rewritten as
d
dtidq =M(R,L)idq −
1
Lu(t) + d(t) (2.13)
with
u(t) = u(t)v(t)
d(t) = d0 +M(R,L)i∗dq −d
dti∗dq.
(2.14)
The ability to steer idq to the origin (i.e. perfectly track the current reference) requires
the compensation of the T-periodic (T = 1/(2πωm)) disturbance term d(t). In order to
robustly cope with this issue, according to standard internal model-based techniques, the
two components of d(t) can be thought as generated by the following exosystem
wi = Ωwi(t), wi ∈ R2N+1, i = d, q
di(t) = Γiwi(t)(2.15)
34
2.4. Input saturation and current bound issues
where Γi ∈ R(1×2N+1) are suitably defined vectors and matrix Ω ∈ R
(2N+1)×(2N+1) is
defined as Ω = blkdiag(Ωj) with Ω0 = 0 and
Ωj =
[
0 jωm
−jωm 0
]
, j = 1, . . . , N. (2.16)
with the pairs (Γi, Ω) observable. Thus, defining Φ = blkdiag(Ω,Ω) and Γ = blkdiag(Γd,Γq),
the following internal model-based controller can be designed
ξ = Φξ +Qidq
u(t) = Γξ +Kidq, u(t) = uuc(t) = u(t)/v(t)(2.17)
where K, Q are free design parameters to be selected to ensure asymptotic stability at
the origin of the following closed-loop error dynamics, resulting from the interconnection
of systems (2.17) and (2.13)
˙idq =
(
M(R,L)− K
L
)
idq + Γξ
˙ξ = Qidq +Φξ
(2.18)
where ξ = ξ − Lw(t). According to the parametrization introduced in [74], K and Q can
be chosen as K = diag(ki), i = d, q, and Q = Π−1GK. Where G = blkdiag(Gi), i = d, q
is composed by column vectors which paired with two arbitrary Hurwitz matrices Fi ∈R(2N+1)×(2N+1) form controllable pairs. While Π = blkdiag(Πi), i = d, q is formed by
the solutions of the following Sylvester equations, that by the observability hypothesis on
(Γi,Ω), are ensured to exist and being non singular
FiΠi −ΠiΩ = −GiΓi, i=d,q. (2.19)
Further details on the tuning of K and Q will be provided in ch. 3, where the stability
properties of the overall unconstrained system error dynamics, involving also the DC-bus
voltage v(t), is analyzed. Here, relying on the existence of such result, which implies
bounded v(t) inside the predefined range, uuc imposed at the SAF input, is assumed
always well-defined (see eq. (2.17)).
2.4 Input saturation and current bound issues
The SAF two main limitations mentioned in 2.1.1, namely a bounded control authority
due to the maximum voltage on the converter legs (that clearly cannot rise beyond the DC-
bus capacitor voltage) and a maximum current value that the filter can provide without
damaging its components, need to be formulated in terms of the input and state variables
of the SAF state-space model derived in (2.8). With this results at hand, the anti-windup
structure in 1.5 can be specialized to this particular application.
First the input constraints are concerned; by (2.3) and applying the transformations de-
fined in (2.5), (2.7) it can be verified that the control input vector u has to lie inside the
35
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
−0.8 −0.4 0 0.4 0.8−0.8
−0.4
0
0.4
0.8
rin
d
q
Figure 2.3: Control input feasibility space.
hexagon reported in Fig.2.3, where each vertex corresponds to a possible discrete config-
uration of the inverter switching network, while all the other points can be obtained as
mean value in a PWM period (see [32] for further details). For the sake of simplicity, and
to guarantee a certain margin, since realistic converters cannot be driven by PWM mod-
ulation indexes uxyz ranging exactly on [0, 1], the hexagon is commonly approximated by
its inscribed circle. Hence the following saturation constraint on the control input vector
is formulated
‖u(t)‖ =‖u(t)‖v(t)
≤ rin =1√3, ∀t (2.20)
where rin is the radius of the inscribed circle.
As far as the converter current limitation is concerned, given the threshold value Imax of
the filter current vector norm, and recalling (2.11), the constraint can be mapped into the
following inequality
‖i∗dq + (η 0)T + idq‖ ≤ Imax (2.21)
where, for reasons that will be clear in the following, the overall current reference i∗dq has
been subdivided into the terms i∗dq == [Ild0 − ild − ilq]T , related to the load harmonics
compensation, and η related to the DC-bus voltage stabilization.
In [71], [32] a filter sizing procedure, which allows to comply with condition (2.20), given
a worst case scenario for the non linear load current profiles that the filter is expected
to compensate for, and nominal three-phase line voltage conditions, has been presented.
In plain words, the capacitor voltage lower bound vm has to be large enough to avoid
saturation for the current controller under a worst load profile scenario. This sizing rule
can guarantee saturation avoidance under perfect tracking of the considered load currents
scenario, whatever the adopted current controller is (it is based on SAF nominal model
inversion). Moreover, depending on the vm oversizing and the adopted current controllers,
saturation can be prevented even when some tracking errors are present. Similarly, the
SAF switching devices can be selected so that the maximum permissible current norm
Imax is large enough to comply with (2.21) under the given worst case load profile. Never-
theless, as previously remarked, SAF are expected to operate also under harsh conditions,
where the margins given by a suitable dimensioning are not sufficient, and the saturation
limits can be temporary or permanently hit.
For what concerns control input saturation, beside initial tracking error, the most common
36
2.4. Input saturation and current bound issues
SAF Parameters
Inductances L [mH] 3
Parasitic resistance R [Ω] 0.12
DC-bus capacitance C [mF] 9.4
Nominal line voltage amplitude V ∗m [V] 310
DC-bus voltage working range [vm, vM ] [V] [700 900]
Maximum filter current vector norm Imax [A] 70
Table 2.1: Shunt Active Filter parameters.
causes for SAFs are mains voltage amplitude fluctuations and too large load currents to
compensate for, i.e.. unfeasible reference for the above defined current controller. Re-
calling equations (2.9), (2.14), it can be noted that the line voltage amplitude and large
current harmonics affect the disturbance term d(t), through the constant component d0
and the oscillatory term M(R,L)i∗dq − ddt i
∗dq, respectively. Then, by (2.17), it can be veri-
fied how this two components acts on the control action, influencing the controller integral
part for what concern d0, and the remaining controller internal states devoted to reproduce
the load currents oscillatory dynamics. Provided that the line voltage disturbance has to
be rejected in order to steer the filter currents for the harmonics compensation, since the
overall control effort is limited, it is straightforward to conclude that too large references
i∗dq cannot be tracked without violating the inequality (2.20). It’s further to notice that
not only the amplitude, but also the angular frequency of the current harmonics affects
the control effort.
Similarly, a sudden rise of the voltage amplitude, and then of d0, would require the most of
the control effort to be put on the integral action, leaving less room to compensate for the
disturbance oscillatory terms. As a result inequality (2.20) can be violated even if i∗dq is
a feasible reference under nominal operating conditions. It’s further to remark that these
considerations are valid for any current control solution, since the ability to asymptotically
track the current harmonics mandates to cope with the rejection of disturbance d(t).
As mentioned in the previous chapters, if not managed properly, control input saturation
produces the well known pernicious windup effects; a strong loss of performance and, in
some cases, stability properties. For the considered application, particular attention has
to be paid to avoid this situations, since each undesired behavior of the filter currents is
reflected to the grid line. In Fig. 2.5, 2.4 the consequences of a 20% mains voltage ampli-
tude increase, in the absence of a suitable anti-windup scheme, are reporteded. Simulations
have been carried out considering a benchmark system characterized by the parameters
reported in 2.1, while two sinusoidal harmonics, with amplitude 5 A at frequency 7ωm
and 13ωm (corresponding respectivly to 6ωm, 12ωm in the d− q rotating reference frame),
have been adopted to reproduce a nonlinear load current profile.
When the line voltage amplitude step occurs (at time t = 1.5 sec.), a strong degradation
of tracking performance, highlighted by the tracking error waveforms in Fig. 2.4, can be
37
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
noted. As a consequence, highly distorted currents are injected on the line grid, as the
main current waveform in Fig. 2.5 shows. Moreover, when the mains voltages amplitude
is restored to the nominal value (at time t = 3 sec.), the windup effect is clearly present on
the control inputs, this enforces strong oscillations on the mains currents for a significant
transient time.
As concern current constraint (2.21), in principle a current reference, related to the non-
linear load according to (2.11), can require an overall control action that lies inside the
admissible set, but it can still be unfeasible because it exceeds the system maximum cur-
rent rating. This typically occurs if the range for the DC-bus voltage is set to high values,
providing a high control authority on the inverter legs, while the filter current limit Imax is
relatively small. When this condition takes place, a suitable reference current saturation
strategy has to be applied. As it will be showed in 3, the active current term η plays
a crucial role, as it prevent the system energy storage device from discharging. For this
reason it is suitable to preserve it, and make the current reference to fulfill (2.21 ) by prop-
erly reshaping i∗dq, possibly with minimal impairment to the compensation performances.
This issue is deeply discussed in 2.6, where a current saturation approach, relying on the
anti-windup unit for control input saturation, described in the next section, is presented.
1.45 1.5 1.55 1.6−15−7.5
07.515
time [s]
i d[A
]
2.95 3 3.05 3.1−60−30
03060
time [s]
i d[A
]
(a) Current error, d-component.
1.45 1.5 1.55 1.6−2.5
−1.250
1.252.5
time [s]
i q[A
]
2.95 3 3.05 3.1−10−5
05
10
time [s]
i q[A
]
(b) Current error, q-component.
1.45 1.5 1.55 1.6−600−300
0300600
time [s]
v ma[V
]
2.95 3 3.05 3.1−600−300
0300600
time [s]
v ma[V
]
(c) Line voltage, phase a.
Figure 2.4: Current tracking performance under voltage amplitude transient: transition
between nominal and saturation condition (on the left) and viceversa (on the
right).
38
2.5. Current control Anti-Windup scheme
1.45 1.5 1.55 1.60246
time [s]
||uuc||
2.95 3 3.05 3.1−15
01530
time [s]
||uuc||
(a) Input vector norm.
1.45 1.5 1.55 1.6−10−5
05
10
time [s]
i ma[A
]
2.95 3 3.05 3.1−30−15
01530
time [s]
i ma[A
](b) Line current, phase a.
1.45 1.5 1.55 1.6−15−7.5
07.515
time [s]
i la[A
]
2.95 3 3.05 3.1−15−7.5
07.515
time [s]
i la[A
]
(c) Load current, phase a.
Figure 2.5: Saturation (on the left) and windup (on the right) effects on control input
norm and mains current.
2.5 Current control Anti-Windup scheme
The system resulting from the interconnection of the SAF current subsystem, obtained by
(2.8) neglecting the DC-bus voltage state equation, and the internal model-based controller
(2.17), matches the system general form reported in (1.55), (1.56). As a consequence the
closed-loop currents error dynamics (2.18) correspond to the general expression given in
(1.57); the current state matrixM(R,L) plays the role of the general function f , while, ac-
cording to (2.20), the admissible control set for the SAF input vector is U = u : ‖u‖ ≤ rinwhich is obviously compact and connected. Moreover, the functional controllability hy-
pothesis are clearly satisfied by the square and bilinear SAF currents subsystem. On
the other hand, it has to be underscored that the DC-link voltage value v(t) affects the
currents tracking error dynamics (2.13), by multiplying the control input (see (2.14)).
However, here a suitable voltage stabilizer is assumed, and, as stated in 2.1.1, the focus
is first put on the current controller, neglecting the internal dynamics stabilization, then
some specific countermeasures will be taken to not impair the voltage controller, whatever
its realization is, with the anti-windup unit.
In view of these considerations, the methodology sketched in 1.5 can be specialized to
design an anti-windup solution for the SAF current subsystem. To this aim, consider
39
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
−0.8
−0.4
0
0. 4
0.8
0.80.404−0.8−0.
q
dUr
d
U
raw
Ur
u
Figure 2.6: Definition of vector dUr .
the following reduced feasibility set, specializing definition (1.61) for the SAF application:
Ur := u : ‖u‖ ≤ r, with r = rin − raw and raw ∈ [0, rin[. Now denote the distance
function dist(Ur, u) from a generic control vector u to the set Ur; exploiting the circular
shape of the input feasibility set, the following projection vector can be defined, according
to the geometric idea described in Fig. 2.6.
dUr(u) =
dist(Ur, u)−∇dist(Ur ,u)‖∇dist(Ur ,u)‖ if u /∈ U
0 if u ∈ U(2.22)
That being defined, the results of the anti-windup procedure in 1.5, applied to the SAF
current control problem, can be summarized in the following proposition.
Proposition 2.5.1 Consider the current subsystem and the corresponding controller re-
spectively reported in the first equation of (2.8) and (2.17). Assume v(t) is always inside
the interval specified by objective O2 in 2.2.
Then, if an anti-windup unit is constructed exploiting the following elements,
• the additional reference dynamics
d
dtiaw = h1SAF (Ur, uuc, i
∗dq, idq, iaw) + h2SAF (·)
h1SAF =M(R,L)iaw − v
LdUr(uuc)
h2SAF = σ(Kaw iaw
), iaw = iaw − iaw
(2.23)
with Kaw an arbitrary Hurwitz matrix and
σ(Kaw iaw
)=
Kaw iaw if ‖LvKaw iaw‖ ≤ raw
Kaw iaw‖Kaw iaw‖raw if ‖L
vKaw iaw‖ > raw(2.24)
iaw =v
LM(R,L)−1dUr(uuc) (2.25)
where uuc is the unconstrained control vector given by the internal model-based law
in (2.17).
40
2.5. Current control Anti-Windup scheme
• the additional feed-forward term (similar to what in (1.59))
gawSAF (·) =L
v
(
M(R,L)iaw − diawdt
)
(2.26)
and, in the controller (2.17) providing the control command uuc, the tracking error idq is
replaced by ˜idq , idq − i∗dq − iaw,
and the overall control action u is re-defined (similar to 1.63) as follows
u = uuc + gawSAF (·) = uuc +L
v
(
M(R,L)iaw − d
dtiaw
)
=
= uuc +
g1SAF︷ ︸︸ ︷
L
v(M(R,L)iaw − h1SAF )−
L
vh2SAF
(2.27)
then the following holds
1. The new tracking error variables ˜idq have the same dynamics as those of the orig-
inal closed-loop error system reported in (2.18), furthermore they are structurally
decoupled from the additional reference dynamics iaw;
2. The additional reference dynamics (2.23) is bounded;
3. The Euclidean norm of the current reference modification ‖iaw‖ tends to decrease
when the control vector uuc (defined by (2.17) with ˜idq replacing idq) belongs to the
admissible set U = u : ‖u‖ ≤ rin;
4. The u(t) re-defined in (2.27) is in U ∀ t.
Proof 1. Consider the overall control action defined in (2.27), replacing it in (2.13),
and expliciting uuc according to (2.17), by direct computation the error dynamics
become˙idq =
(
M(R,L)− 1
LK
)
˜idq + Γξ
˙ξ = Φξ +Q˜idq
(2.28)
This system is completely decoupled from the reference modification dynamics in
(2.23), moreover it is identical with the original tracking error dynamics idq given
by (2.18). Therefore, it’s easy to guess that the stability properties of the original
tracking dynamics are preserved, and no modification of the unconstrained feedback
controller is required.
2. First note that the distance vector dUr(uuc) is norm bounded, since the disturbance
term d(t) lumping togheter the load harmonics and line voltage effects, is obviously
bounded, and (2.28) is decoupled from (2.23) and GAS (according to the design of
the unconstrained controller). Furthermore, the term h2SAF , being the output of a
saturation function, is bounded by definition, and the matrixM(R,L) is Hurwitz.As
a result system (2.23) is a stable linear system driven by the bounded input dUr(uuc),
hence boundedness of the trajectories iaw(t) trivially follows.
41
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
3. From definition (2.22) and (2.25), it is straightforward to verify that when uuc ∈ U ,then dU (uuc) and iaw are null. Now consider the Lyapunov candidate V (iaw) =
‖iaw‖2, taking its derivative along the system (2.23) trajectories under the above
mentioned conditions (i.e. dU (uuc) = 0, iaw = iaw) yields
d
dt‖iaw‖22 = −2R‖iaw‖22 + iTawσ(Kawiaw) < 0
where negative definiteness of V (iaw) clearly stems from the properties of the ma-
trices M(R,L), Kaw, and the function σ(·) defined in (2.24) which, as it can easily
verified, preserves the sign of its input argument.
4. By the definitions of h1SAF , h2SAF in (2.27) the overall control action in (2.23) can
be written as
u(t) = uuc(t) + dUr(uuc)−L
vσ(Kaw iaw) (2.29)
therefore, according to definitions (2.22), (2.24), and the obvious requirement raw ∈[0, rin[, the following inequality holds
‖u(t)‖ ≤ ‖uuc + dUr(uuc)‖+∥∥∥∥−Lvσ(Kaw iaw)
∥∥∥∥≤ r + raw ≤ rin (2.30)
which shows that the re-defined u(t) is always in U .
Remark It’s further to notice that by the choice of h1SAF , we obtain g1SAF , defined in
(2.27) similarly to that in (1.63), equal to the distance vector dUr . Hence, recalling the
considerations reported in 1.5, g1SAF is actually the minimum norm vector that steers the
overall control action inside the set Ur. In other words uuc + g1SAF is enforced to lie on
the boundary of the restricted admissible set ∂Ur for any possible input saturation sce-
nario. Therefore no control authority, except for the part reserved to shape the additional
reference dynamics by means of h2SAF , is lost for anti-windup purposes.
Remark From (2.23), it can be verified that iaw defined in (2.25) corresponds to the
constant iaw (i.e.. with null derivative) steady state reference modification value, that
would be required to prevent saturation under constant saturation scenarios. Where for
constant saturation scenario is meant a working condition producing a constant (on both
amplitude and direction) vector dUr(uuc). Computing the corresponding error dynamics,
by (2.23) and (2.25) yields
d
dtiaw =M(R,L)iaw + σ(Kaw iaw). (2.31)
Similarly to what in item 3 of the above proof, by simple Lyapunov arguments and the
properties ofM(R,L),Kaw and σ(·), it’s straightforward to verify that, in these conditions,
the origin is an asymptotically stable equilibrium point of (2.31), i.e. iaw asymptotically
approaches iaw.
In this particular application, the considered anti-windup additional dynamics would be
42
2.5. Current control Anti-Windup scheme
stable even without the shaping term h2SAF , as it replicates the plant free dynamics,
which is given by the Hurwitz matrix M(R,L). On the other hand, the anti-windup
convergence property would depend only on the system electrical parameters (specifically
on the ratio R/L), which can give rise to poorly damped dynamics. As a result the
actual reference modification iaw could reach iaw after a long transient phase during which
additional spurious currents could be injected into the system, impairing the compensation
performance. Therefore the shaping term h2SAF = σ(Kaw iaw) is introduced to endow the
anti-windup dynamics with the desired convergence rate and comply with objective c)
stated in 1.5. As a certain part of the control action, specifically the annulus of thickness
raw, needs to be reserved for the above mentioned stabilizing action, a suitable trade-off
between the anti-windup dynamics convergence properties and the loss of control authority
has to be sought.
By the last remark it’s clear that, under constant saturation conditions, the anti-windup
unit would achieve objective c) stated in 1.5, since a constant reference modification, that
in the considered synchronous coordinates corresponds to oscillating currents at the line
frequency, would be injected, without introducing additional harmonics. While under
different scenarios, beside stability and saturation avoidance would still be ensured, the
requirement to not inject spurious currents by means of the anti-windup system could
be violated. Moreover, under realistic operating condition, a constant steady state iaw
is hardly reached. In fact, according to the internal model principle, the steady state
unconstrained vector, has to reproduce the output of the exosystem 2.15 to ensure perfect
tracking, i.e. by 2.13 uuc = Lv d(t). Thus the control action is clearly influenced by the
load harmonics oscillations, this also affects the distance vector dUr(uuc) which in turn
acts on iaw through (2.25). In conclusion the anti-windup unit so far designed could inject
additional electrical pollution into the mains.
These considerations are confirmed by simulation tests carried out under the same sce-
nario reported in 2.4. In Figs. 2.7, 2.8 it can be noticed that, albeit the input vector
saturation is prevented, the current error variables are not affected by the line voltage
step, and a smooth transition between saturation and nominal conditions,with no bumps
or windup effects, is ensured, the harmonic compensation performances are very poor
during saturation condition, as showed by the mains current waveform in Fig. 2.8(c) and
the corresponding magnitude spectrum in Fig. 2.8(d). For the sake of simplicity the sim-
ulation test have been carried out with no stabilizing action in the anti-windup system,
that is Kaw and raw have been set to null values. However, the introduction of the term
h2SAF would not improve the system behavior in terms of compensation performances, in
fact, as previously remarked, the effects of such a stabilizing action are mainly devoted to
steer the anti-windup dynamics towards a stationary (in this case constant) steady-state
reference modification iaw.
Another issue stemming from the straightforward application of the generic anti-windup
strategy in 1.5 to the SAF current controller, is that the reference modification can act on
both the active and reactive current components. In view of what previously mentioned,
43
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
1.45 1.5 1.55 1.60
0.20.40.6
time [s]‖u
‖
2.95 3 3.05 3.10
0.20.40.6
time [s]
‖u‖
1.45 1.5 1.55 1.6−10−5
05
10
time [s]
i ma[A
]
2.95 3 3.05 3.1−10−5
05
10
time [s]
i ma[A
]
Figure 2.7: Control input norm (saturation limit in red) and main current waveforms with
anti-windup solution implemented: transition between nominal and saturation
condition (left), and viceversa (on the right).
as it will be deeply elaborated in 3, the first influences relevantly the DC-bus power flow,
perturbing the voltage stabilizer action.
2.5.1 Improvements in the anti-windup strategy
In order to overcome the drawbacks arising from the straightforward application of the
general anti-windup strategy to the SAF system, and to achieve objectives b) and c) stated
in 1.5, the anti-windup unit design is modified exploiting some general insights combined
with the SAF structural properties. The first step is to generate a reference modification
acting only on the q-component of the current reference, so that the anti-windup unit is, as
much as possible, decoupled from the DC-link voltage dynamics, and, as a consequence,
it will minimally impair the voltage stabilizer action during saturation. To make the
anti-windup dynamics (2.23) compliant with these restriction on the additional reference
current components, we define the new distance vector
dUr(u) =
dist[−ωmL R](Ur, u)−∇dist[−ωmL R](Ur,u)
‖∇dist[−ωmL R](Ur,u)‖ if u /∈ U0 if u ∈ U
(2.32)
where dist[−ωmL R](·) denotes the distance from the generic vector u to the set Ur along
the direction defined by the vector [−ωmL R]T (see Fig. 2.9 for the geometrical represen-
tation). Then h1SAF is redefined accordingly, replacing dUr(u) with dUr(u) in (2.23). It
can be verified that this variation generates a steady stave value iaw (defines as in 2.25)
which has the d-component structurally null, i.e. iaw = [0 iawq]T .
The next modification to guarantee a proper harmonic cancellation under saturated con-
ditions is to make iawq “as constant as possible”, despite of the oscillations related to
current harmonics to be compensated for. For this purpose the following almost constant
44
2.5. Current control Anti-Windup scheme
1.45 1.5 1.55 1.6−0.01
−0.0050
0.0050.01
time [s]
˜ i d[A
]
2.95 3 3.05 3.1−0.01
−0.0050
0.0050.01
time [s]
˜ i d[A
]
(a) Current error d-component
1.45 1.5 1.55 1.6−0.01
−0.0050
0.0050.01
time [s]
˜ i q[A
]
2.95 3 3.05 3.1−0.01
−0.0050
0.0050.01
time [s]
˜ i q[A
]
(b) Current error q-component
1.45 1.5 1.55 1.6−10−5
05
10
time [s]
i ma[A
]
2.95 3 3.05 3.1−10−5
05
10
time [s]
i ma[A
]
(c) Mains current, phase a
0 200 400 600 800 10000
3.25
6.5
frequency [Hz]
i la[A
]
0 200 400 600 800 10000
3.25
6.5
frequency [Hz]
i ma[A
]
(d) Load and mains current magnitude spectrum.
Figure 2.8: Current tracking performance with anti-windup solution implemented: tran-
sition between nominal and saturation condition (on the left) and viceversa
(on the right).
distance vector is adopted in place of dUr(uuc)
dUr(uuc) = dUr(uuc(arg maxτ∈]t−T,t]
(||dUr(uuc(τ))||))). (2.33)
this selection takes advantage from T-periodicity of steady-state control signals needed
for the perfect harmonics tracking, in order to define a constant (or piecewise constant
in case of no stationary load profiles) distance vector. In this way, bearing in mind the
previous remarks, even for realistic saturation scenarios, the anti-windup unit defined in
Prop. 2.5.1 will asymptotically approach a steady state constant (or piecewise constant)
value defined by¯iaw = [0 ¯iawq]
T =v
LM−1dUr(uuc). (2.34)
On the other hand, due to this choice the term g1SAF will no longer be minimal, in the
sense reported in 1.5; indeed during saturation, the modified control vector u will often
range in the interior of the set Ur instead of lying exactly on the its boundary. Even if in
45
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
−0. 8
−0.4
0
0.4
0.8
0.80 0.4−0.4−0. 8
uuc
dUr
u
d
q
Ur
Figure 2.9: Definition of vector dUr .
this case the control action compatible with the constraints and the shaping of the anti-
windup dynamics would not be fully exploited, since the anti windup scheme produces a
constant reference modification, no other harmonics will be injected into the system, and
the compensation performance of the system will be significantly improved.
Owing to the considered restriction on the form of the reference modification iaw, some
feasibility issues could arise with respect to the maximum operative region enlargement
objective a stated in 1.5, this topic will be carefully analyzed in the next Section.
2.6 Dealing with current and anti-windup unit limitations
The anti-windup solution presented in 2.5 introduces an additional current term to prevent
control input saturation, however, as mentioned in 2.4, shunt active filters are subject also
to a maximum current limitation. Furthermore the modifications introduced in 2.5.1 to act
on the solely current q-component, in order not to affect the voltage stabilizer performance,
reduces the degrees of freedom available to the anti-windup unit. As a consequence the
set of operating conditions that can be handled is reduced.
This two issues can be addressed by limiting the current reference term i∗dq related to the
load current harmonics. In particular a proper reference scaling need to be performed so
that the anti-windup unit is ensured to have the authority to enforce the overall control
action inside the admissible set, by means of a reference modification iaw compliant with
the filter current limitation. Obviously the procedure has account also for the other
variables appearing in (2.21), i.e. the term η, exploited to stabilize the DC-link voltage
dynamics, and the current tracking error idq which also affects the control action through
the stabilizing output feedback terms in (2.17).
In the following, first an optimization problem will be formulated to evaluate what are the
maximum current references that can be tracked without violating the system constraints.
In this respect it will be showed how the anti-windup unit, defined in 2.5, actually provides
a significant extension of the system operative range. Then, based on the maximum
reference estimation, a suitable current saturation strategy will be introduced to cope
with unfeasible load currents.
46
2.6. Dealing with current and anti-windup unit limitations
−0. 8
−0. 4
0
0.4
0.8
−0. 8 −0. 4 0 0.4 0.8
Ur
d
q
Figure 2.10: Feasibility space when an AW unit acting on iq is considered.
2.6.1 General problem formulation
When the proposed anti-windup unit is added to the system, the current constraint in-
equality (2.21) is modified as
‖Cx∗ + idq(t) +¯iaw + iaw(t) + (0 η)T ‖ ≤ Imax (2.35)
where, in order to underscore the contribution of each harmonic component, the reference
term i∗dq has been expressed as the output of the following system
x∗ = Sx∗, i∗dq = Cx∗
S = blkdiag(Si), Si =
[
0 ωi
−ωi0
]
, C =
[
1 0 1 . . .
0 1 0 . . .
]
, i = 1, . . . , N.(2.36)
As regards the control input feasibility set, by considering the fact that the anti-windup
reference modification is allowed to act only on the current q-component, which in turn
induces an additional feed-forward action (2.26) acting along the direction given by the
vector [ωL − R]T , we obtain that the set of unconstrained control actions that can be
steered inside the feasibility space Ur := uuc ≤ r by adding a reference term in the form
reported in (2.34), is the region between the two half-planes (see Fig. 2.10)
uTuc
[R
ωmL1
]T
≤√(
1 +R
ωmL
)2
r2
uTuc
[R
ωmL1
]T
≥ −√(
1 +R
ωmL
)2
r2
(2.37)
Recalling (2.15) (2.14) and(2.36), by simple computations, the unconstrained control ac-
tion uuc provided by the internal model-based controller (2.17) can be written as
uuc =Γξ + L
[M(Cx∗ + (η 0)T )− CSx∗ − (0 η)T + d0
]+Kidq
v. (2.38)
Then, by inequality (2.37) and Fig. 2.10, it’s obvious that the proposed anti-windup
unit enlarges the range of the admissible unconstrained control input vectors which, if no
anti-windup unit was introduced, would coincide with the inscribed circle in Fig. 2.3. On
the other hand, the constraint to have a null d-component on the reference modification
47
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
current vector ¯iaw causes the control vector directions lying outside the strip in Fig. 2.10
to be unrecoverable, in the sense that they cannot be led inside the admissible circle by the
anti-windup system. This issue would be avoided if the anti-windup reference signal was
allowed to have a non null d-component, however, as discussed in 2.5, this solution is not
suitable to keep a proper voltage dynamics behavior also under saturation, as requested by
objective b) in 2.1.1. Moreover, the input constraint has to be combined with the current
limitation; roughly speaking, no matter the form of the anti-windup reference signal, if
the unconstrained control vector norm is too large, the required current modification can
violate inequality (2.35).
As mentioned, here the objective is to formulate a suitable optimization problem to eval-
uate the maximum amplitude of the current harmonics, collected in the vector x∗, that
the filter can compensate for, without exceeding the current and control action constraints
expressed in (2.37), (2.35) respectively. To this aim, the following optimization variables
are considered: x∗, idq, ξ, η, η, iaw, iaw, while the line grid amplitude Vm and the DC-link
voltage v(t) are regarded as problem parameters.
In order to obtain meaningful results, we need to add some restrictions regarding the error
variables idq, ξ, along with the current term η, and its derivative η. As regards η, a con-
servative bound on the maximum value necessary to keep the DC-link voltage inside the
admissible range can be estimated, under the reasonable assumptions that the the voltage
initial condition belongs to this range, and the system power losses, for which, as it will
be shown in 3.1, it has to compensate for, are bounded. Similarly a bound can be set on
the maximum derivative η. Thus the following additional constraints are introduced
|η| ≤ ηmax, η| ≤ ηmax. (2.39)
It’s further to remark that the DC-bus voltage controller has to be suitably saturated so
that the actual output η satisfies the above limitations.
For what concerns the error variables, we assume that, under a worst case scenario, the
initial conditions χ(0) = [idq(0) ξ(0)]T and iaw(0) ranges respectively on finite regions W0
and I0. In view of these considerations, the largest feasible current reference i∗dq = Cx∗
can be computed as the solution of the following problem
maxx∗ ,idq ,ξ,η,η,iaw, ¯
awi‖Cx∗‖
subject to (2.35), (2.37), (2.39), (2.18), (2.23)
∀ t > 0, ∀ χ(0) ∈ W0, ∀ iaw(0) ∈ I0.
(2.40)
2.6.2 Reduced problem formulation
Problem (2.40) is highly nonlinear and strongly interlaced, in the sense that the constraints
equations depends on several optimization variables, via involved functions, e.g the dis-
tance function defined in Fig. 2.9. Moreover the constraints depends on time, even if we
can exploit periodicity of the current harmonics and consider just a single line grid period,
we need to solve semi-infinite problems.
48
2.6. Dealing with current and anti-windup unit limitations
−0.02 −0.01 0 0.01 0.02
−0.02
−0.01
0
0.01
0.02
rη
q
d
(a) Control action devoted to
DC-bus voltage stabilization
−0.8
−0.4
0
0.4
0.8
−0.8 −0.4 0 0.4 0.8
rηd
q
(b) Reduced feasibility set af-
ter computation of rη.
−0. 8 −0. 4 0 0. 4 0. 8−0. 8
−0. 4
0
0. 4
0. 8
rη d
q
(c) Reduced feasibility set (in
red) obtained subtracting Pη.
Figure 2.11: Approximation of the control term devoted to track the current reference
component η.
In order to obtain a numerically tractable version of the optimization problem, a sort
of “clusterization” can be performed; namely the contributions that the optimization
variables give to the the current and control action constraints are separately consid-
ered. Then, a reduced problem, involving only the reference harmonics vector x∗ and
the anti-windup signal ¯iaw as optimization variables is formulated, by bounding the terms
depending on the other original optimization variables and subtracting the obtained ap-
proximated sets to the original constraints inequalities.
Starting with the current term η, by (2.38) the steady-state control effort required to
perfectly track it is
uη =L
vdc(M [η 0]T − [0 η]) (2.41)
hence constraints in (2.39) can be mapped to uη, obtaining the polytope (see Fig. 2.11(a))
Pη := coM [±ηmax 0]T − [0 ± ηmax]
, where co denotes the convex hull of the vectors
representing the polygon vertices. Then, for the sake of simplicity, the polytope with
is approximated with its circumscribed circle of radius rη, obtaining a norm constraint
‖uη‖ ≤ rη. Finally this control effort can be subtracted to uuc in (2.38) (see Fig. Fig.
2.11(b)); by this procedure the control action authority required to track η and then sta-
bilize the DC-link voltage, is preserved given a worst case scenario. This, combined with
the choice to act only on the q-component reference for anti-windup purposes, allows to
completely decouple the anti-windup and the DC-bus capacitor voltage dynamics. As a
result the voltage controller performance is not impaired by the anti-windup unit, and
objective b) in 1.5 is fulfilled. In principle conservatism can be reduced by directly sub-
tracting the polytope P instead of its circumscribed circle to the set Ur . The obtained
set is reported in Fig. 2.11(c) where it can be seen that the approximation error made
taking the inscribed circle is reasonably small.
Now we move to estimate the control effort related to the error variables χ = [idq ξ]T
that, by (2.38), results
˜u =[K Γ]χ
v. (2.42)
49
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
Recalling that the initial condition χ(0) is assumed to belong to the region W0 the goal is
to find a tight bound on the trajectories of system (2.18) that for convenience are rewritten
as
˙χ = Aχ, with A =
[
M(R,L)− KL Γ
Q Φ
]
(2.43)
A rather simple method to bound the error trajectories is by means of quadratic Lyapunov
functions ellipsoidal level sets ([75], [23]). In particular, given a Lyapunov candidate in
the form V (χ) = χTPχ, associated to (2.43), the smallest invariant ellipsoid E(P ) :=χ : χTPχ ≤ 1
enclosing W0 would provide a bound for the trajectories χ(t), and, as
direct consequence on the size of ‖χ(t)‖.By the invariance hypothesis on E(P ) it follows ‖χ(t)‖ ≤ maxχ∈E(P ) ‖χ(t)‖, therefore the
peak value of the error vector can be expressed as
α :=√
max ‖χ‖ : χTPχ ≤ 1. (2.44)
Now consider the setχ : ‖χ‖ ≤ α2
which can be also regarded as the scaled unit ball
E( Iα2 , 1) :=
χ : χT Iχα2 ≤ 1
. Therefore, it’s easy to verify that α ≥ α if E(P, 1) ⊂ E( Iα2 ).
In this respect, α can be equivalently defined as
α = min
α : P ≥ I
α2
(2.45)
thus the best upper bound of ‖χ‖ can be computed minimizing α. In view of these
considerations, solving the problem
minP>0,δ
δ
s.t.
[
P I
I δI
]
≥ 0
W0 ⊂ E(P )ATP + PA < 0
(2.46)
provides an upper bound√δ∗ on the peak value of χ and a symmetric positive definite
matrix P ∗ defining the smallest invariant ellipsoid containing W0. Note that the first
constraint is equivalent to inequality (2.45) by Schur’s complement, while the second and
third constraints enforce the invariance of E(P ) for each χ(0) ∈ W0. When the set W0 is a
polytope, i.e. W0 := co χvi , i = 1, . . . , 24N+4, by convexity the set inclusion W0 ⊂ E(P )can be expressed as a linear inequality in the variable P , hence the above problem is cast
into the following eigenvalue problem
minP>0,δ
δ
s.t. χTviPχvi ≤ 1, i = 1, . . . , 24N+4
[
P I
I δI
]
≥ 0
ATP + PA < 0
(2.47)
50
2.6. Dealing with current and anti-windup unit limitations
−0.6 −0.3 0 0.3 0.6−0.6
−0.3
0
0.3
0.6
Ur
Ur rη + β
Figure 2.12: Vector ˆuuc feasibility space.
in practical conditions, this situation arises when the initial values of the error vector
components χi(0), i = 1, . . . , 4N + 4 are decoupled and known to belong to a range
[χimin, χimax]. Finally, by (2.42), it is straightforward to verify that the norm of ˜u is
ensured to be less then β = 1v(t)
√
δ∗λmax([K Γ]T [K Γ]). Hence, in order to completely
bound the control authority related to all the terms that don’t depend directly on the
harmonics vector x∗, also β, along with rη, is subtracted to the overall control action. As
a result, an equivalent feasibility set involving the control action ˆuuc =Lv ((M−CS)x∗+d0),
explicitly depending on the current harmonics, can be considered by replacing uuc with
ˆuuc and r with ˆr = r − rη − β in (2.37). The obtained set is shown in Fig. 2.12.
A similar procedure can be adopted to deal with the current limitation (2.35); beside the
term η the terms idq, iawneed to be bounded. As regards the tracking error idq, since it is
part of the vector χ, which, by the previous analysis, is ensured to belong to the ellipsoid
E(P ∗) ∀ t, its size can be evaluated by projecting the invariant ellipsoid onto the plane
id, iq. In order to consider a worst case condition, the largest projection has to be sought.
To this aim first the extremal values of id, iq belonging to the invariant boundary surface
are computed by solving the following convex optimization problems
max(min)id(iq)
s.t. χP ∗χ ≤ 1.(2.48)
then the resulting R2 ellipsoid P ∗
idefined by the optimal values of (2.48) is approximated
by its circumscribed circle of radius ri =1√
λmin(P ∗
i)to finally obtain the bound ‖idq‖ ≤ ri.
As concerns the anti-windup unit error variables iaw(t), the same analysis carried out to
handle the error vector χ can be repeated, i.e. by solving
minQ>0,δaw
δaw
s.t. I0 ⊂ E(Q)[
Q I
I δawI
]
≥ 0
(M(R,L) +Kaw)TQ+Q(M(R,L) +Kaw) < 0
(2.49)
51
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
−0. 8 −0. 4 0 0. 4 0. 8−0. 8
−0. 4
0
0. 4
0. 8
. . .
. .. P1P2
P3P4
P5
P6
F
dF
Figure 2.13: Control inputs constraints combined with AW current limitation.
an upper bound√δ∗aw for ‖iaw(t)‖ is obtained. Summarizing these results, and recalling
(2.34) constraint (2.35) can be approximated as
‖Cx∗‖ − |iawq| ≤ Imax (2.50)
with Imax = Imax − |ηmax| − ri −√δ∗aw.
Even with the adopted reduction, the maximum current and control input constraints are
interlaced, since ¯iawq in (2.50) depends on ˆuuc through (2.34). In other words, the feasible
control input vectors ¯uuc are those that lie in the region between the half-planes (defined
similarly to what in 2.37) shown in Fig. 2.12 and can be led inside the feasible set Ur :=‖ˆuuc‖ ≤ ˆr
by means of an anti-windup reference modification ¯iaw fulfilling inequality
(2.50). By (2.29) it can be verified that the anti-windup unit action is geometrically
equivalent to translating the feasibility circle by ±LvM(R,L)(‖iaw‖). Summarizing all
these considerations, for a given ¯iaw, the feasibility set F reported in Fig. 2.13 is obtained.
Therefore, since the line grid voltage disturbance d0 has to be compensated to ensure
system stability, bearing in mind the previous reasoning, by (2.38) we can state that a
current reference vector i∗dq = Cx∗ is certainly feasible if it requires a corresponding control
action ˆu∗uc such that
‖ˆu∗uc‖ =
∥∥∥∥
L
v(M − CS)x∗
∥∥∥∥≤ dF (iawq) (2.51)
where dF
(¯iawq
)
= dist(u∗, ∂F) is the distance from the point u∗ =(Vm
v , 0)to the
boundary ∂F of the feasibility set (see Fig. 2.13). Exploiting geometrical considerations,
dF (iawq) can be computed as the optimal value of the following problem
minˆu∗uc,λ≥0
∥∥∥∥∥ˆu∗uc −
(Vmv
0
)T∥∥∥∥∥
∇ˆu∗uc
∥∥∥∥∥ˆu∗uc −
(Vmv
0
)T∥∥∥∥∥+ λ∇ˆu∗
ucg(ˆu∗uc,
¯iaw) = 0
g(ˆu∗uc,¯iawq) = 0
(2.52)
where, according to the Lagrange multipliers theory, the first constraint is a sort of tan-
gency condition between the circle containing the control action devoted to track the
52
2.6. Dealing with current and anti-windup unit limitations
current harmonics, and the boundary of the feasible set ∂F which is represented by the
piecewise function
g =
ˆu∗Tuc [R
wmL 1]T − q P1d ≤ ˆu∗ucd ≤ P2d, ˆu∗ucq ≤ P2q
ˆu∗Tuc [R
wmL 1]T + q P3d ˆu∗ucd ≤ P4d, ˆu
∗ucq ≤ P3q
‖ˆu∗uc − gSAF ‖2 − r2 ˆu∗ucd ≥ P2d or P4d ≤ ˆu∗ucd ≤ P2d, ˆu∗ucq ≤ P6q
‖ˆu∗uc − gSAF ‖2 − r2 u∗ucd ≥ P3d or P1d ≤ ˆu∗ucd ≤ P3d, ˆu∗ucq ≥ P5q
(2.53)
where q =
√(
1 + RωmL
)2ˆr2. Clearly dF depends on the anti-windup current modification
¯iawq, more precisely it can be verified that ddiawq
dF ≥ 0, i.e. a larger |iawq| would enlarge
the left and right boundaries of F . Before formulating the final reduce problem, inequality
(2.51) is rearranged as
L
v
√√√√
N∑
i=1
c2i x∗2i ≤ dF
ci =√
λmax((M − Si)T (M − Si))
(2.54)
where x∗i is the amplitude of the ith current harmonic x∗i . Note that in (2.54) we consider
a worst case condition where all the control terms ˆuuci =Lv (M − Si)x∗i related to each
harmonic are aligned in the same direction. In a similar fashion current constraint (2.50)
is expressed as√√√√
N∑
i=1
x∗2i + |iawq| ≤ Imax. (2.55)
eventually the following reduced optimization problem is derived
maxx∗1 ,iawq
√√√√
N∑
i=1
k2i x∗1
s.t.L
v
√√√√
N∑
i=1
k2i c2i x
∗1 ≤ dF (iawq)
√√√√
N∑
i=1
k2i x∗1 + |iawq| ≤ Imax, x
∗1 ≥ 0.
(2.56)
where dF is computed as the optimal value of problem (2.52), and, assuming a known load
current spectrum, the harmonics components have been expressed in terms of the lowest
frequency harmonics x∗1 to be compensated for, by means of the (known) gains ki.
The outlined procedure is based on a suitable clusterization of the terms involved in the
original nonlinear optimization problem (2.40), this leads in principle to a conservative
solution. Moreover the sets bounding each group of variables are approximated by their
circumscribed spherical regions. However the problem complexity have been fairly reduced.
In this respect, note that for a given of dF (iawq), problem (2.56) is a linear programming
53
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
problem. Therefore the computation of the maximum feasible current reference is reduced
to solve a sequence of LP problems for different values of ¯iawq and, in turn of dF (iawq)
coming from the solution of (2.52). Hence the problem can be quite solved by means of
pretty standard active set or interior point methods available for example inMATLABTM
fmincon constrained optimization function.
Bearing in mind the considerations on how ¯iawq influences the size of the control input
feasibility set, and by standard linear programming arguments, it is easy to guess that the
optimal value ¯i∗awq of (2.56) satisfies
Imax − |i∗awq|√∑N
i k2i
=dF
Lv
√∑N
i=1 c2i k
2i
(2.57)
that is the control input and current constraints are both active at the optimal point.
2.7 Current saturation strategy
Problem (2.56) provides an upper bound on the amplitude of the load current harmonics
that can be compensated for, adopting the proposed anti-windup strategy. By solving it for
the instantaneous DC-link voltage v(t) and line voltage amplitude Vm values, the maximum
size for the reference term i∗dq to be feasible can be estimated as: ‖i∗dqmax‖ =∑N
i=1 kix1.
Whith this result at hand the current reference derived by the load currents, as reported
in (2.11), can be shaped to fulfill the system constraints, according to the following law
i∗dqsat = αi∗dq, α = sat10(α∗)
α∗ = minτ∈[t,t−T [
‖i∗dqmax‖(Vm(t), v(t))
i∗dq(τ)
(2.58)
where sat10(·) denotes a scalar saturation function with bounds [0, 1]. By this saturation
strategy, provided that the assumptions on the current tracking error initial values and
the term η made in 2.6 are satisfied, the anti-windup unit has always the authority to steer
the control input vector inside the feasible set without incurring into current limitation
problems. Also in this case T-periodicity in the load currents has been exploited, in or-
der to produce a constant (or piecewise constant for time-varying load profiles) reference
scaling factor α∗. Thus, frequent and abrupt reference bumps that would be impossible
to track by the internal model based current controller are avoided.
By (2.54) and (2.36) it’s straightforward to verify that the harmonics frequency affects
control input constraints through the term ci, and that higher frequency harmonics de-
mand a larger control effort to be compensated for. Hence, assuming known frequencies
for the load profile, in order to ensure feasibility for any possible scenario, i∗dqmax has to
be computed by solving problem (2.56) for a current reference entirely generated by the
highest frequency load current component, i.e. setting ki = 1 for i = N and ki = 0 other-
wise.
Due to this choice, and the worst case scenario approximations made in 2.6 to bound
54
2.8. Numerical and simulation results
the contributions of the error and DC-link stabilization variables, the above saturation
strategy can lead to significantly conservative results. In principle conservatism could be
reduced by exploiting the available measures of the current error variables, and the actual
values of the variables η and ¯iawq. Based on these informations, the computation of the
scaling factor of α∗ in (2.58) can be replaced by the following more “speculative” strategy
α∗ = minτ∈[t,t−T [
Imax − ‖idq(t)‖ − ‖iaw(t)‖ − |η| − |iawq|i∗dq(τ)
(2.59)
however, in general, no formal guarantee of current reference feasibility would be provided
by the above law.
2.8 Numerical and simulation results
In order to confirm the effectiveness of the considered SAF saturated control strategy,
extensive simulation tests have been carried out considering a realistic filter characterized
by the parameters reported in Tab. 2.1. In this respect, first a numerical comparison
between the maximum reference achievable with the proposed saturated control strategy
and the one obtained with no anti-windup solution has been carried out. The results are
discussed in the next paragraph.
2.8.1 Numerical results
Consider the SAF parameters reported in Tab. 2.1, and set the DC-link voltage and
line grid voltage amplitude respectively to the lower end of the admissible working range
v = vm and the nominal value Vm = V ∗m. As regards the variable η the limit values
are: ηmax = 10A, ˙ηmax = 10A/s, while the error variables initial conditions idq, ξ iaw
are assumed to range in: idqi ∈ [−20, 20]A, i = d, q, ξi ∈ [−10, 10]A, i = 1, . . . , 4N + 2,
iaw ∈ [−10, 10]A i = d, q. Finally a margin raw = 0.05 is reserved for the anti-windup
stabilizing action h2SAF . If no anti-windup augmentation is performed, the same analysis
made in (2.6.2) to derive problem (2.56) can be carried out to formulate the problem
maxx∗1 ,iawq
√√√√
N∑
i=1
k2i x∗1
s.t.L
v
√√√√
N∑
i=1
k2i c2i x
∗1 ≤ ˆrnoAW
√√√√
N∑
i=1
k2i x∗1 ≤ Imax, x
∗1 ≥ 0.
(2.60)
where Imax = Imax − ri − |ηmax| and ˆrnoAW = rin − rη − β since no anti-windup unit is
implemented. The considered benchmark load profile consists of the first two harmonics of
a three-phase diode rectifiers (a common nonlinear load in industrial plants) that, in the
55
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
Maximum current comparison between AW and no AW solutions
AW no AW
x∗1 (300 [Hz]) 17.67 [A] 7.99 [A]
x∗2 (600 [Hz]) 7.35 [A] 3.32 [A]¯i∗awq 25.6 [A] n.a
Table 2.2
d−q reference frame, are placed at 6ωm, 12ωm. The amplitude ratio is set according to the
rectifier spectrum (see [67]). The numerical values of the optimal variables obtained solving
(2.56) and (2.60) with the above defined parameters, are reported in Tab. 2.2. As stated
in 2.7, in order to guarantee the saturation law (2.58) to effectively tackle all the possible
load scenarios, given the frequency spectrum, problems (2.60) ,(2.56) need to be solved
for a single harmonics at 12fm; in this scenario we obtain respectively ‖i∗dqmax‖ = 6.67 A
and ‖i∗dqmax‖ = 11.81 A with ¯i∗awq = 30.1A. It’s worth to notice that almost half of the
available current is used by the anti-windup unit, this high current request is due to the
fact that only the q component is exploited for anti-windup purposes, inducing a constraint
on the direction of the feed-forward action as reported in Fig. 2.9. In section 2.9 possible
improvements to overcome this issue will be discussed, however, how it will be showed in
the next paragraph, this approach is suitable to cope with practical cases of abrupt line
voltage amplitude variations and load currents that would produce input saturation if no
anti-windup was implemented.
2.8.2 Simulations
The first scenario to be reproduced is the same reported in 2.4 and 2.5, in order to confirm
the effectiveness of the improvements reported in 2.5.1. The matrix Kaw in (2.23) is set
to diag(kaw), with kaw = −5R/L, while the norm of the corresponding feed-forward termLvKaw iaw is saturated to raw = 0.05. Figures 2.14, 2.15 show the obtained results; the
tracking error behavior is similar to that obtained with the anti-windup dynamics design
carried out in 2.5, while, as regards the harmonics compensation, when the mains voltage
amplitude disturbance is increased, the cancellation performance are not relevantly im-
paired. In fact, thanks to the adopted improvements in the anti-windup dynamics design,
only the constant term ¯iawq, corresponding to an inductive current term, is injected into
the mains by the anti-windup unit, while the two disturbances harmonics are effectively
compensated, also during saturation conditions. The FFT of the load and main currents
reported in Fig. 2.15(c) confirms this fact. The effects of the stabilizing action Kaw iaw
can be noted in Fig. 2.14(a) where the control action waveform reaches the limit value rin
during the anti-windup dynamics transient, then, when the steady state value is reached,
it lies strictly below the limit according to the defined margin raw. As a result the can-
cellation performances are improved, since the constant steady state is quickly reached,
56
2.8. Numerical and simulation results
1.45 1.5 1.55 1.60
0.20.40.6
time [s]
||u||
2.95 3 3.05 3.10
0.20.40.6
time [s]
||u||
(a) Input vector norm.
1.45 1.5 1.55 1.6−10−5
05
10
time [s]
i ma[A
]
2.95 3 3.05 3.1−10−5
05
10
time [s]
i ma[A
](b) Line current, phase a.
1.45 1.5 1.55 1.6−600−300
0300600
time [s]
v ma[V
]
2.95 3 3.05 3.1−600−300
0300600
time [s]
v ma[V
]
(c) Line voltage, phase a.
Figure 2.14: Control input norm (limit in red) and mains current under line voltage steps
with improved anti-windup solution.
and the effects of the anti-windup dynamics transient are barely noticed at the line side
as showed by the waveform of the line current ima in Fig. 2.14(b). It is worth to remark
that the anti-windup scheme, combined with the current saturation strategy defined in 2.7,
can handle an instantaneous line voltage amplitude increase of arbitrary value, provided
that the required current ¯iawq is below Imax.
A second simulation scenario has been set to prove the scheme effectiveness to handle load
current harmonics that would be unfeasible without the anti-windup augmentation. For
this purpose, the load current profile, selected as the first two harmonics of a three-phase
AC-DC rectifier (composed by a diode bridge), is switched between a feasible scenario and
the maximum values computed in 2.8.1; the obtained results are reported in Figs. 2.16,
2.17. Also in this case without anti-windup solutions the current tracking performance
drastically degrade, while the anti-windup unit is able to exploit the available current
margin to introduce a current reference modification on the q component that prevents
the control input vector saturation. By the magnitude spectrum of the line current ima
reported in Fig. 2.17(c), it can be verified that the adopted strategy allows to totally
compensate for the increased load current, thus enlarging the set of trajectories that the
current controller can effectively track.
57
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
1.45 1.5 1.55 1.6−0.01
−0.0050
0.0050.01
time [s]
˜ i d[A
]
2.95 3 3.05 3.1−0.01
−0.0050
0.0050.01
time [s]˜ i d
[A]
(a) Current error, d-component.
1.45 1.5 1.55 1.6−0.01
−0.0050
0.0050.01
time [s]
˜ i q[A
]
2.95 3 3.05 3.1−0.01
−0.0050
0.0050.01
time [s]
˜ i q[A
]
(b) Current error, q-component.
0 200 400 600 800 10000
3.25
6.5
frequency [Hz]
i la[A
]
0 200 400 600 800 10000
3.25
6.5
frequency [Hz]
i ma[A
]
(c) Magnitude spectrum of load and main currents under
saturation.
Figure 2.15: Current tracking performance under line voltage steps with improved anti-
windup solution: transition between nominal and saturation condition (on
the left) and viceversa (on the right).
58
2.8. Numerical and simulation results
−30 −15 0 15 30−30
0
30
˜id [A]
˜ i q[A
]
0 200 400 600 800 10000
5
10
frequency [Hz]
i ma[A
]
(a) Current error phase portrait and load current magni-
tude spectrum.
1.45 1.5 1.55 1.60
1
2
time [s]
||u||
2.95 3 3.05 3.10
1
2
time [s]
||u||
(b) Input vector norm.
1.45 1.5 1.55 1.6−10−50510
time [s]
i ma[A
]
2.95 3 3.05 3.1−20−10
01020
time [s]
i ma[A
]
(c) Line current, phase a.
1.45 1.5 1.55 1.6−30−15
01530
time [s]
i la[A
]
2.95 3 3.05 3.1−30−15
01530
time [s]
i la[A
]
(d) Load current, phase a.
Figure 2.16: System behavior under large current harmonics with no anti-windup.
59
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
−0.01 −0.005 0 0.005 0.01−0.015
0
0.015
˜iq [A]
˜ i q[A
]
0 200 400 600 800 10000
25
50
frequency [Hz]
i ma[A
]
(a) Current error and load current magnitude spectrum.
1.45 1.5 1.55 1.60
0.20.40.6
time [s]
||u||
2.95 3 3.05 3.10
0.20.40.6
time [s]
||u||
(b) Input vector norm.
1.45 1.5 1.55 1.6−30−15
01530
time [s]
i ma[A
]
2.95 3 3.05 3.1−30−15
01530
time [s]
i ma[A
]
(c) Line current, phase a.
1.45 1.5 1.55 1.6−30−15
01530
time [s]
i la[A
]
2.95 3 3.05 3.1−30−15
01530
time [s]
i la[A
]
(d) Load current, phase a.
Figure 2.17: System behavior under large current harmonics with anti-windup solution.
60
2.9. Alternative SAF AW unit design
Furthermore, even if the current tracking is momentary lost due to the reference disconti-
nuity produced by the load switch, in contrast with the case with no anti-windup augmen-
tation, the current error variables are quickly steered to zero. The different behaviors are
highlighted by the current error variables phase portrait reported in Fig. 2.16(a), 2.17(a).
In this set of simulations the less conservative current saturation strategy (2.59) has been
adopted in order to show the effectiveness of the control input anti-windup scheme under
the maximum reference currents calculated in 2.8.1.
2.9 Alternative SAF AW unit design
InSubsection 2.5.1 the focus has been put on designing a filter anti-windup unit producing
a constant reference modification acting on the current q-component in order to achieve
all the objective stated in 2.1.1. On the other hand this choice entails a limitation on
the input feasibility set enlargement; as showed in 2.6, all the input vector lying outside
the region between the two half-planes defined by (2.37) can not be led back to the
feasible circle U with such a reference modification form. In principle this drawback can
be overcome by enriching the set of possible reference modifications the anti-windup unit
can generate. Owing to requirement c) in 2.1.1, it turns out that, along with constants,
the other only possible choice is to inject current signals at the same frequency of the
load current harmonics. In this respect, based on the load current measures ildq, the load
current harmonics xl can be estimated by a standard Luenberger observer
xl = Sxl + L(Cx− ildq) (2.61)
with S, C defined as in (2.36). Now assume the vector uuc lies outside the region between
the half-planes in (2.37), a possible solution is to compute the minimum radius R1 ≥ rin :
∃ ¯iaw such that ‖uuc + gawSAF ‖ = R1 (see Fig. 2.18(a)), and steer the unconstrained
vector on UR1 := u : ‖u‖ ≤ R1 by means of (2.23), then augment the anti-windup unit
with the following dynamics
d
dtiaw1 = h1SAF1(Ur, uuc + gawSAF , i
∗dq, idq, iaw1) + h2SAF1(·)
h1SAF1 =M(R,L)iaw1 −v
Ld1Ur
(uuc + gawSAF )
h2SAF1 = σ(Kaw1iaw1
), iaw1 = iaw1 − iaw1
(2.62)
with
iaw1 =v
L
λCxl,
λ = − v
Ldist[MC−CS]x(Ur, uuc + gawSAF ). (2.63)
Similarly to what in (2.23) dist[MC−CS]x(·) denotes the distance taken along the direction
given by [MC−CS]x, and d1Ur(·) the corresponding vector defined similarly to (2.32). Also
in this case a stabilizing action can be introduced and properly saturated on the circle
of radius raw1 (preventively subtracted to U along with raw) by means of the function
σ(·) in (2.24). The above dynamics has to be combined with the feed-forward action
gaw1SAF = Lv
(M(R,L)iaws − diaws
dt
), so that the redefined overall control action u =
61
Chapter 2. SATURATED NONLINEAR CURRENT CONTROL OF SHUNT ACTIVE FILTERS
uuc + gawSAF + gaw1SAF is inside U . After some computations it can be proved that
dynamics (2.62) is endowed with all the properties stated in Prop. 2.5.1, and that iaws
is the oscillatory steady state to which iaws is steered by the property of M(R,L) and
the stabilizing action Kawsiaws1. Roughly speaking, recalling (2.11), a scaled version of
the term i∗dq is subtracted to the overall current reference, so that the input constraint is
fulfilled.
However, even adding dynamics (2.63) does not ensure feasibility for all the possible
directions of uuc, in particular during the anti-windup dynamics transient, when iaw1 6=iaw1. Assuming that uuc+gawSAF cannot be enforced in U by acting only on the direction
given by (MC−CS)xl (see Fig. 2.18(b)), we can iterate the approach presented before; i.e.
after computing the minimum radius R2 : ∃ ¯iaw1 such that ‖uuc+gawSAF+gawsSAF ‖ = R2,
the vector uuc + gawSAF is steerd into UR2 by (2.63), and the following additional anti-
windup dynamics is introduced
d
dtiaw2 = h1SAF2(Ur, uuc + gawSAF + gaw1SAF , i
∗dq, idq, iaw2) + h2SAF2(·)
h1SAF2 =M(R,L)iaw2 −v
Ld2Ur
(uuc + gawSAF + gaw1SAF )
h2SAF2 = σ(Kaw2iaw2
), iaw2 = iaw2 − iaw2
(2.64)
withiaw2 =
v
L(M − Ω2)
−1d2Ur(uuc + gawSAF + gaw1SAF )
Ω2 =
[
0 ω2
−ω2 0
](2.65)
and ω2 and arbitrary angular frequency, obviously different from the load harmonic fre-
quencies. As usual a stabilizing action h2SAF2 has been added also for this part of the
anti-windup dynamics, and a proper control authority margin raw2 has to be preserved
for it. Here d2Ur(uuc + gawSAF + gaw1SAF ), defined similarly to what in (2.32), denotes the
distance vector from uuc+gawSAF +gaw1SAF to Ur taken along the vector [(MC−CS)x]⊥,which defines an orthogonal direction with respect to (MC−CS)x (see Fig. 2.18(b)). Re-
placing (2.65) into (2.64) it’s easy to verify that iaw2 is the steady-state sine-wave signal, at
frequency ω2, to which dynamics (2.64) is steered thanks to the properties of the matrices
M −Ω2 and Kaw2. Furthermore, the same steps reported in the proof of Proposition 2.5.1
can be repeated to state the same results for (2.65). Finally, adding the corresponding
feed-forward action gaw2SAF = Lv
(
M(R,L)iaw2 − diaw2dt
)
, applying the same reasoning re-
ported in 2.5, it can be verified that uuc + gawSAF + gaw1SAF + gaw2SAF ∈ U .It’s worth noticing that, the shaping of the current reference is already included in dy-
namics (2.62) through the term λ, hence the current saturation strategy can be lumped
together with the computation of the anti-windup reference signals ¯iaw, iaw1, iaw2. For
this purpose we express d2Ur(uuc + gawSAF + gaw1SAF ) as
d2Ur(uuc + gawSAF + gaw1SAF ) = γ
[(MC − CS)xl]⊥
‖[(MC − CS)xl]⊥‖β = distUr(uuc + gawSAF + gaw1SAF )
(2.66)
62
2.9. Alternative SAF AW unit design
−0.8
−0.4
0
0.4
0.8
−0.8 −0.4 0 0.4 0.8
uuc
gawSAFd1
Ur
UR1
d
q
(a) Effects of the anti-windup
term iaw1.
−0.8
−0.4
0
0.4
0.8
−0.8 −0.4 0 0.4 0.8
gaw1SAF
d2Ur
UR2
d
q
(b) Effects of the anti-
windup term iaw2.
Figure 2.18: Feasibility set enlargement by anti-windup dynamics enrichment.
then, combining (2.34), (2.63), (2.65), the current constraint inequality (2.21) is modified
as‖i∗dq + (η 0)T + idq +
¯iaw + iaw +v
LλCxl+
+ iaw1 +v
L(M − Ω2)
−1γ[(MC − CS)x]⊥ + iaw2‖ ≤ Imax
(2.67)
with
γ =β
‖[(MC − CS)xl]⊥‖.
Bearing in mind all the previous considerations, the following optimization problem can
be formulated in order to comply with the SAF input and maximum current constraints
minλ,β,iaw
β
uuc + gawSAF + gaw1SAF + gaw2SAF ∈ U‖i∗dq + (η 0)T + idq +
¯iaw + iaw +v
LλCxl + iaw1+
+v
L(M − Ω2)
−1β[(MC − CS)x]⊥ + iaw2‖ ≤ Imax
(2.68)
This alternative seems promising, since it allows to enlarge the range of control actions
that can be steered inside the feasible set to all the possible directions in R2, without
affecting the DC − bus voltage dynamics, since no active current terms (i.e. constant
d-component) are injected by the anti-windup dynamics. Thus the only limitation in the
filter operating range would be due to the constraints on the drained current. On the
other hand, due to the several control and reference terms added by the anti-windup unit,
reducing (2.68) to a numerically tractable problem in a similar fashion to what carried out
in 2.6.2 is a much harder task. Moreover the term iaw2 actually introduces a steady-state
spurious current signal which, in general, is not ensured to vanish when uuc+gawSAF comes
back to the range that can be compensated with the solely dynamics (2.62). However,
in the future this solution could be further explored to enhance the results presented in
2.8.2.
63
Chapter 3
On the control of DC-link voltage
in Shunt Active Filters
In this chapter the stabilization of the SAF non-minimum phase zero dynamics,
given by the Shunt active filter DC-bus voltage, is addressed. The issue is mo-
tivated by the physics of the system, showing how the capacitor would inevitably
discharge if no compensation action is applied, then, relying on a suitable sys-
tem dimensioning, providing a frequency separation between the voltage and filter
currents dynamics, two different control solutions are analyzed.
3.1 Problem statement
In ch. 2 the issue of the DC-bus voltage non-minimum phase behavior has been sketched,
mentioning how part of the current reference η needs to be devoted to keep the DC-bus
voltage value inside the range defined by O2 in 2.2.2, in a backstepping fashion typical for
underactuated systems. Before presenting the possible solutions to provide η by a voltage
stabilizer reported in Fig. 2.2, a formal motivation of the non minimum phase behavior
is stated. In this respect, consider the ideal reference term i∗dq = [ild − ild0 ilq]T , defined
in (2.11), then the first equation in (2.8) can be rewritten as
u(t) = uv(t) =M(R,L)
(
i∗dq + idq −di∗dqdt
− didqdt
+ d0
)
(3.1)
thus, it turns out that the steady state voltage dynamics, corresponding to a perfect
tracking of i∗dq is
v2 = ǫL
(
d0 +M(R,L)i∗dq(t)−d
dti∗dq(t)
)T
i∗dq. (3.2)
The integrator is driven by two periodic signals, with period T = 1/fm: the zero mean
value component ǫL(d0− ddt i
∗)T i∗dq, and the signal ǫL(M(R,L)i∗dq)T i∗dq which has negative
mean value as long as parasitic resistance R or reference i∗dq are not null. By this, even if
the initial voltage value of the DC-link is inside the desired range, it will leave it in finite
64
3.1. Problem statement
time, providing a loss of controllability of the system due to the capacitor discharge.
To avoid this phenomenon, the reference must be revised, taking into account an additional
active current term, which should be drained from the line grid by the active filter, in order
to compensate for its power losses. Following this motivation, the current reference signal
should be modified as i∗dqϕ0= i∗dq + (ϕ0 0)T , where ϕ0 is the active current needed to
compensate for the power losses, namely the value that would make the signal driving
the integrator in (3.2) with zero mean value. Albeit ϕ0 can be in principle computed by
solving the following power balancing equation ([32],[33])
Rϕ20 − Vmϕ0 +Rfm
∫ 1/fm
0(i∗2d (τ) + i∗2q (τ))dτ = 0 (3.3)
due to the system parameters uncertainties, its value is in general unknown or heavily
inaccurate. Hence it needs to be reconstructed by means of η, by a suitable elaboration
of the DC-bus voltage measures. To this purpose, considering the change of variables
z = v2(t) − V 2∗, where V 2∗ is the square desired reference voltage value (usually set
to (v2M + v2m)/2), and recalling (2.13), (2.8), the overall system error dynamics can be
expressed asd
dtidq =M(R,L)idq −
v
Lu(t) + d(t)
˙z = ǫuT [idq + i∗dq + (η 0)T ].
(3.4)
Therefore objective O2 can be equivalently formulated in the error variable z requiring
z(t) ∈ [−l∗, l∗] for all t ≥ t0, with l∗ = (v2M −v2m)/2,provided that z0 ∈ [−l∗, l∗]. It is worth
remarking that, the hypothesis in O2, to start the filter operation with the DC-bus voltage
already inside the admissible range is not limiting, since, due to the L − C resonance ad
the free-wheeling diodes, the natural uncontrolled response of a three-phase AC-DC boost
converter brings the DC-bus voltage to a value that is twice the line peak-to-peak voltage,
which is in general grater then the lower bound vm. Thus the controller can be switched
after an initial free response transient phase, having z(t0) ∈ [−l∗, l∗].In summary the goal of a voltage stabilizer is to steer, by means of the virtual input η,
z towards a steady state where the voltage trajectories are free to oscillate within the
admissible region, but their mean value is null. In this chapter two possible approaches
to achieve this goal are discussed. In Section 3.2, a preliminary control oriented capacitor
sizing procedure is discussed; by (3.4) it can be clearly noted that the capacitor value affects
the amplitude of the voltage oscillations through the term ǫ defined in (2.9), therefore, no
matter what is the adopted control solution to generate η, if the capacitor is undersized,
its voltage oscillations will extend beyond the desired range [vm, vM ].
However, a suitable capacitor sizing can be exploited also to induce a time-scale separation
between the filter currents and DC-bus voltage dynamics; with this result at hand, the
voltage stabilization problem can be carried out separately from the current tracking
controller, then the overall system practical stability can be formally stated by means
of singular perturbation and input to state stability results for two-time scales systems
([76]). Relying on such design procedure, in Section 3.3 a robust integral control solution,
65
Chapter 3. ON THE CONTROL OF DC-LINK VOLTAGE IN SHUNT ACTIVE FILTERS
proposed in [70] to asymptotically recover the unknown power losses related term ϕ0 is
briefly presented. In Section 3.4, an averaged control solution, carried out in the so-
called phasor’s domain as proposed in [33], and able to minimize the effects of the voltage
stabilization on the harmonic cancellation performance, is discussed, providing a complete
analysis of a possible real-time control implementation.
3.2 Control oriented DC-Bus capacitor sizing
As mentioned in 2.2.2, the SAF current tracking problem and DC-bus voltage stabilization
are interlaced, since the ability to steer the filter currents relies on the energy stored in the
DC-bus capacitor, and, in turn, the capacitor voltage depends on the currents absorbed
/delivered to the mains. In order to tackle the two problems by separate controllers, a sort
of frequency separation between the dynamics of the SAF current and voltage subsystems
has to be induced. To this aim, noting that system (2.8) can be viewed as a singular
perturbation model, parametrized by ǫ, and then, by (2.9), on C, a suitable capacitor
sizing has to be carried out. Moreover, as previously mentioned , the voltage on the DC-
bus is enforced to oscillate during current harmonics compensation, therefore, if objective
O2 has to be fulfilled, the capacitor size has to be large enough to prevent the voltage
oscillations to exit the predefined range ((see (3.2), (3.4)).
A possible procedure ([32], [71]) to cope with this issues is the following; determine the
maximum energy that the capacitor has to exchange over a line period T by solving, in
the variables z consisting in the filter current harmonics (2N +1) magnitudes and 2N +1
phases, the problem
Emax = maxz
maxt
∣∣∣∣
∫ t
t0
[Vm 0]T idq(τ)dτ
∣∣∣∣
(3.5)
subject to the constraints
• switches currents must be less than the maximum rating
• the control output must be feasible, i.e. u ∈ U
• harmonics components phases have to be greater than −π and less than π.
Then, assuming the voltage variation corresponding to Emax is V ∗ − vm, where vm is the
DC-bus voltage range lower bound, the capacitor value design equation can be written as
C =2Emax
V ∗2 − v2m(3.6)
this sizing rule ensures the voltage oscillations are bounded inside the range [vm, vM ] for
a considered worst case scenario, and in general it is suitable to provide a time-scale
separation between the current and voltage subsystems.
66
3.3. Robust integral control of voltage dynamics
3.3 Robust integral control of voltage dynamics
A possible approach to deal with the DC-bus voltage dynamics stabilization was presented
in [70], [31], the proposed control structure to generate η is
η = Nq(z) +Nθ
θ = −ǫh(z)(3.7)
where N = (1 0)T , q(·) the deadzone function
q(z) =
0 if z < l
z − zsign(z) otherwise(3.8)
with l ≤ l∗ and h(·) a differentiable function satisfying h(z) = 0 ∀z : |z| ≤ l. This control
law can be motivated as follows: the role of θ is to introduce an integral action in the
voltage dynamics, with the aim to estimate the unknown term ϕ0. Since h(·) is null if
|z| ≤ l, the integral term is inactive if the voltage value lies strictly inside [vm, vM ], while if
v(t) approaches the boundary of the admissible range, the function h(·) has to be chosen so
that θ approaches ϕ0. In addition, when z approaches the admissible bounds, the integral
action is enriched with a further stabilizing term q(z). Roughly speaking, the rationale
of this control solution is to make the voltage controller to minimally interfere with the
current controller, precisely only when the boundaries of a suitable deadzone function are
approached by z, so that the harmonics cancellation performance are maximized. Here
the stability proof of the overall system under the internal model based current control
presented in 2.3 and the stabilizer (3.7) is sketched for the sake of completeness. Replacing
(2.17) and (3.7) into (3.4) the overall closed-loop dynamics in the error variables becomes
˙idq =
(
M(R,L)− 1
LK
)
idq −1
LΓξ + I(z, θ, ˙z,
˙θ)
˙ξ = Qidq +Φξ
˙z = ǫ(LdT (t)N(θ − q(z))) + γ1(·) + γ2(·) + de(t)˙θ = −ǫh(z)
(3.9)
where θ = θ − ϕ0, and γ1(idq, ξ, z, θ) = (Γξ +Kidq)(idq −Nq(z) +Nθ), γ2(idq, ξ) = (Γξ +
Kidq)(i∗dq+Nϕ0)+Ld
T (t)idq, de(t) = LdT (t)(i∗dq+Nϕ0), and I(z, θ, ˙z,˙θ) =M(R,L)N(θ−
q(z)) − N(˙θ − dq(z)
dt
)
. Then, relying on a suitable dimensioning procedure, briefly de-
scribed in (3.2), we can assume ǫ to be “small enough” so that the two time-scale averaging
theory (see [15], [76]) can be applied to system (3.9). Namely the overall dynamics can
be viewed as the interconnection between the fast subsystem (χ = [idq ξ]) and the slow
subsystem (z, θ). It is further to notice that ǫ has been introduced in the integral action of
the controller (3.7) in order to keep the voltage controller speed in scale with the voltage
subsystem, thus maintaining the two-time scale behavior of the closed-loop system.
In accordance with the general singular perturbation theory, the effectiveness of the con-
trol law (2.17), can be proved by considering the boundary layer system, obtained taking
67
Chapter 3. ON THE CONTROL OF DC-LINK VOLTAGE IN SHUNT ACTIVE FILTERS
ǫ = 0 in (3.9), which gives
˙idq =
(
M(R,L)− 1
LK
)
idq −1
LΓξ + I(z, θ, 0, 0)
˙ξ = Qidq +Φξ.
(3.10)
If the matrices K and Q are selected as in 2.3, on the basis of the solution of (2.19),
asymptoic stability of the above boundary layer system can be stated. In order to prove
this claim, define the vector
Rξ =
[
− R
Γd102N − ωmL
Γq102N
]T
(3.11)
where Γd1, Γq1 denote the first element of vectors Γd, Γq composing matrix Γ in (2.17)
and 02N is a zero raw vector having dimension 2N . Consider now the change of variables
ζ = Πξ −ΠRξ(θ − q(z)) + LGidq (3.12)
where Π = blkdiag(Πd,Πq), G = blkdiag(Gd, Gq). By (2.19), expressing system (3.10) in
this set of coordinates yields
˙i =
(
M(R,L)− 1
LK + ΓL−1G
)
idq −1
LΓΠ−1ζ
˙ζ = F ζ − L(FG−GM(R,L))idq
(3.13)
where F = blkdiag(Fd, Fq). Using standard linear system tools it can be verified that
a value k exists, such that ∀ k ≥ k the matrix K defined as in (2.17) ensures the state
matrix of (3.13) is Hurwitz, hence asymptotic stability of the boundary layer system can
be stated (see [70] for further details).
As regards the slow voltage subsystem (z,θ), a sort of reduced averaged dynamics can be
considered, by confusing, as usual, the fast dynamics with the boundary layer steady state.
Noting that the term de(t) in (3.9) is with zero mean value, and after some computation
(see [31]) the following reduced system is obtained
˙z = −ǫc(q(z)− θ)− ǫR(q(z)− θ)2
˙θ = −ǫh(z)
(3.14)
with c = Vm − 2Rϕ0. Analyzing (3.14) is straightforward to verify that c > 0 is cru-
cial for the system stability, this condition is actually verified by realistic system imple-
mentations. Indeed, from a physical viewpoint, it is reasonable to assume small system
power losses on the parasitic resistances R, i.e. Rϕ0 < Vm. Therefore, it can be proved
that a suitable choice of the function h(·) allows local asymptotic stability of the set
Az :=
(z, θ) : |z| ≤ l, θ = 0
, for the reduced system (3.14). The main result is sum-
marized in the next lemma
Lemma 3.3.1 Consider system (3.14) with
h(z) = ρdq(z)
dzq(z) (3.15)
68
3.4. Averaged control solution
with ρ > 0. Define the compact set Hz(lz) :=
(z, θ) : dist(Az, (z, θ)) ≤ lz
, there exist
l∗z and a class KL function β(·, ·) such that, for all lz ≤ l∗z , the trajectories of (3.14)
originating from Hz(lz) satisfy
dist(Az, (z(t), θ(t))) ≤ β(dist(Az, (z(0), θ(0))), t) (3.16)
Combining the stability results of the boundary layer and reduced dynamics, practical
stability of the set(χ ∈ R
4N+4 : χ = 0× Az naturally stems from Theorem 1 in [76].
This claim is precisely stated in the next proposition [31].
Proposition 3.3.2 Consider the controller (2.17), (3.7) with h(·) selected as in (3.15).
Let Hf ⊂ R4N+4 be an arbitrary compact set and Hz(lz) the set defined in 3.3.1. There
exist positive numbers M , λ, l∗z , k, a class KL function β(·, ·), and, ∀ ν > 0, an ǫ∗ > 0
such that, for all positive lz ≤ l∗z , k > k, and ǫ ≤ ǫ∗, the trajectories of the closed loop
system (3.9), with initial conditions χ(0) ∈ Hf , (z(0), θ(0)) ∈ Hz(lz), are bounded and
satisfy
‖(idq(t) ξ −Rξ(θ(t)− q(z(t)))‖ ≤Me−λt‖idq(0) ξ(0)−Rξ(θ(0)− q(z(0)))‖+ ν (3.17)
with Rξ defined as in (3.11).
It is worth remarking that practical stability result is semi-global for what concern the
fast variables (idq, ξ), while only local as far as the voltage dynamics error variables (z, θ)
are concerned. However this is not an issue for z(0), since, as mentioned, by the theory of
AC/DC boost converters the natural system response steers the DC-bus capacitor voltage
to a level that is twice the mains voltage amplitude. Therefore, relying on a proper system
dimensioning, z(0) can be always assumed inside the desired range. As regards ˜θ(0), as
for realistic implementations ϕ0 is usually very small (the power losses need to be limited
for obvious practical reasons), in practice, setting θ(0) = 0 always fulfills the restriction
on the voltage dynamics initial state.
3.4 Averaged control solution
The solution presented in 3.3 seems suitable to stabilize the DC-bus voltage dynamics
without significantly perturbing the harmonic compensation. However, by a deep analysis
of the slow subsystem dynamics (3.14), it is possible to show that ([70]), under the law
(3.7), the time intervals between two consecutive contacts of z with the boundaries of the
admissible range, are monotonically increasing. In other words, this means that z, and
thus v(t), would approach a steady state value inside the desired range, but close to one
of its bound. As a result, the controller fails to lead the DC-bus voltage mean value to
the desired reference V ∗ = (vM + vm)/2.
In order to overcome this limitation, a possible solution, presented in [33], is to directly
act on the average value of the DC-bus voltage. In this respect the voltage dynamics are
averaged according to the procedure presented in [77], where harmonic analysis is used
69
Chapter 3. ON THE CONTROL OF DC-LINK VOLTAGE IN SHUNT ACTIVE FILTERS
to reduce the system equations of power converters to the dynamics of single harmonics,
so-called phasors.
Following this reasoning, the controlled variable is chosen to be the time-window averaged
value za of the square voltage error z, and the averaging is performed over the time interval
[t−T, t]. In terms of [77], this average value can be regarded as a zero-order phasor defined
as
za(t) =
∫ t
t−Tz(τ)dτ (3.18)
the fact that za is a zero-order phasor, allows to obtain its derivative by simply applying
the same averaging procedure to its differential equation in (3.4)
˙za =1
T
∫ t
t−T
˙z(τ)dτ =ǫL
T
∫ t
t−T
(
M(R,L)i∗dq −di∗dqdt
+ d0
)T
i∗dqdτ + ǫLD(idq) (3.19)
where u has been replaced according to (3.1) and the complete reference idq∗ = i∗dq+(η 0)T ,
while D(idq) collects all the terms depending on the current tracking error idq. The above
average voltage derivative can also be expressed as the difference over one line period of
the actual voltage, hence
d
dt(za) =
d
dt
∫ t
t−Tz(τ)dτ =
z(t)− z(t− T )
T(3.20)
this property connotes the availability of za for measurement in real time, and it is of
crucial importance to actually implement the averaged controller.
The next step is to exploit T-periodicity of the current reference term i∗dq (see also (2.36)),
therefore, when averaged on its own period, a constant value is obtained. Indeed this is
the main motivation to carry out the voltage controller design considering the averaged
dynamics (3.19). In particular, the T-periodic signals in (3.19) can be collected in the
following term
D∗ =1
T
∫ t
t−T
(
M(R,L)i∗dq −di∗dqdt
+ d0
)T
i∗dq
dτ (3.21)
which, bearing in mind the previous considerations, can be regarded as a constant dis-
turbance. Furthermore, since D∗includes the components associated to the system power
losses through the parasitic resistance R; by physical considerations it follows D∗ < 0.
For further simplification the integral operator can be applied to the occurring derivative
terms. By definitions given in (2.11) and (2.9), after some computations, the averaged
error voltage dynamics can be fully expressed in phasor variables, obtaining
˙za = ǫ[Vmηa − 2Rνa − Lνa + LD∗ + LD] (3.22)
where the following nonlinear term has been defined
ν(t) = η(t)
(1
2η(t) + i∗d
)
(3.23)
70
3.4. Averaged control solution
which enters (3.22) with its average and averaged derivative
νa(t) =1
T
∫ t
t−Tν(τ)dτ, νa(t) =
νa(t)− νa(t− T )
T. (3.24)
Hence the averaged voltage trajectories can be steered by means of the averaged control
input
ηa(t) =1
T
∫ t
t−Tη(τ)dτ. (3.25)
Also in this case, a two time-scale behavior of the system can be assumed, relying on a
suitable capacitor sizing that makes ǫ small, and, as a result, enforces the voltage subsys-
tem to be much slower with respect to the current dynamics. Therefore, as in (3.3), the
voltage controller design can be carried out considering only the reduced dynamics, ob-
tained confusing the value of idq with its steady state value idq = 0. It can be verified that
D(0) = 0, then the reduced voltage dynamics is obtained by (3.22) simply dropping the
coupling term D. However, the nonlinear terms νa, and νa cannot be easily managed; be-
side non linearity they contain an integral, a time delay and a time-varying term i∗d(t). In
order to simplify the mathematical treatment, a sort of linearized version of system (3.22)
can be considered. This approximation can be motivated by the following fact ([33]); since
the parasitic resistance R and the filter inductance value L are usually much smaller with
respect to the line voltage amplitude Vm, the nonlinear terms in (3.22) are expected to be
negligible with respect to the the linear ones. In addition, as the current reference active
component i∗d is T-periodic with zero mean value, it does not affect the averaging at all,
as long as η is constant. When η is time-varying, its oscillatory part will be filtered by the
averaging procedure. As a result of the previous steps and considerations, the linearized
reduced averaged model for the DC-bus voltage dynamics can be expressed as
˙za = ǫVm[ηa − ϕ0] (3.26)
this dynamics are considered for control purposes, in particular a standard PI regulator is
selected to produce the control input η, namely
ηa = −kpza + θ
θ = −ǫkiza.(3.27)
The closed-loop system resulting by the interconnection of (3.27) and (3.26), and perform-
ing the change of coordinates θ = θ − ϕ0, results
[˙za˙θ
]
= ǫ
[
−Vmkp Vm
−ki 0
][
za
θ
]
(3.28)
since ǫ and Vm are positive, the state matrix in (3.28) is Hurwitz for any positive gains
kp, ki, thus asymptotic stability at the origin of the above system trivially follows.
Albeit this solution is able to steer the DC-bus voltage average exactly to the desired
reference value V ∗, namely za → 0 while z is free to oscillate according to the load profile
to be compensate for, the resulting control signal ηa is expressed in the space of phasors.
71
Chapter 3. ON THE CONTROL OF DC-LINK VOLTAGE IN SHUNT ACTIVE FILTERS
Therefore a procedure is required to synthesize a real-world control signal η(t) whose mean
value is equal to ηa produced by (3.27). In this respect, note that the derivative of ηa
d
dtηa =
d
dt
1
T
∫ t
t−Tη(τ)dτ (3.29)
can be equivalently expressed on the left side as the difference over one period, while the
right hand side is replaced with what in (3.27)
1
T[η(t)− η(t− T )] = −kp ˙za + ˙
θ = −kp ˙za − ǫkiza (3.30)
then, solving for η(t) yields
η(t) = −Tkp ˙za(t)− ǫTkiza(t) + η(t− T ). (3.31)
The above real time law is actually implementable, since by (3.20), the derivative of the
averaged square voltage error is available from measurements. However, even if stability of
the voltage subsystem would be ensured in the sense of the average value, a further step is
required. In fact, in the incremental implementation (3.31) the integral action is no longer
present, instead, the control input history is kept in memory for one period. Roughly
speaking the current control action is a modification of the value applied at t − T . Con-
sider now that, for the phasor variables system, a stable steady-state guarantees that all
the values assume a constant average, while in principle the real world values can oscillate
freely. This property is desired for what concern the capacitor voltage and, as mentioned,
it is the main motivation for applying the averaging procedure. On the other hand imple-
mentation according to (3.31) can introduce undesired periodic oscillations in the control
input η, which, being remembered through the time delay term, will permanently persist.
In summary, while ηa will approach ϕ0, the actual input η might be any periodic signal
with average value equal to ϕ0. Recalling that η modifies the current reference value i∗d,
any oscillation will impair the harmonics compensation performance.
A possible countermeasure to cope with this issue is to add to (3.31) the following term
dη(t) = η(t− T )− ηa(t− T/2) (3.32)
whose rationale is to correct the stored signal η(t−T ) towards its mean value ηa(t−T/2).It’s worth to remark that, the mean value of the stored signal η(t − T ), is expressed as
its time shifted average value since, in the considered phasor space, the averaged value
does not the corresponds to the actual mean value of its corresponding signal, which is
commonly defined as
sm =1
T
∫ t+T/2
t−T/2s(τ)dτ (3.33)
and is identical to the zero-order phasor definition, except for a time shift of T/2. Note
also that the mean value of the stored signal η(t − T ) can be computed in real time,
because also its “future” values are available. Therefore, the final implementation of the
control law is obtained by (3.31), (3.32)
η(t) = Tηa + ηa(t− T/2) = −Tkp ˙za − ǫTkiza + ηa(t− T/2). (3.34)
72
3.4. Averaged control solution
A further analysis is required to verify that the properties of the averaged voltage system
are preserved under the modification (3.32). This means that the average value ηa, of the
signal η(t) in (3.34) has to approach a steady state equal to the output produced by the
PI averaged law in (3.27). In this respect, define the mismatch variable with respect to
the output of the PI controller in (3.27) ηa = ηa − ηa, by (3.26), (3.27), and (3.34), after
some computations it turns out
˙ηa = ˙ηa − ηa =η(t)− η(t− T )
T− ηa = ηa +
ηa(t− T
2
)− η(t− T )
T− ηa =
=−kpza
(t− T
2
)+ θ
(t− T
2
)+ ηa
(t− T
2
)
T+
−T ηa(t− T )− ηa(t− 3T
2
)
T= . . .
=−kpza
(t− T
2
)+ θ
(t− T
2
)+ ηa
(t− T
2
)
T+ ǫkI za(t− T )+
+ ǫkpVm
(
−kpza(t− T ) + θ(t− T ) + ηa(t− T ))
+kpza
(t− 3T
2
)− θ
(t− 3T
2
)− ηa
(t− 3T
2
)
T
.
(3.35)
Applying the Laplace transform to the previous dynamics yields the following frequency
domain representation
sηa = e−sT2
(ηa − kpza + θ
T+ e−sT
2
(
ǫkiza + ǫkpVm
(
− kpza + θ + ηa
)
+ e−sT2
(kpza − θ − ηa
T
))) (3.36)
which, expressing also (3.26) and (3.27) in the frequency domain, reads as
sηa = e−sT2
(1
T
(
1− kpǫVms
s2 + kpǫVms+ ǫ2kiVm− ǫ2Vmkis2 + kpǫVms+ ǫ2kiVm
)
+
+ e−sT2
(
+ kpǫVm − (kpǫVm)2s
s2 + kpǫVms+ ǫ2kiVm− kpkiǫ
3V 2m
s2 + kpǫVms+ ǫ2kiVm+
+ǫ2kiVms
s2 + kpǫVms+ ǫ2kiVm+ e−sT
2
(1
T
(
− 1 + kpǫVms
s2 + kpǫVms+ ǫ2kiVm+
+ǫ2Vmki
s2 + kpǫVms+ ǫ2kiVm
))))
ηa
(3.37)
thus, drawing the corresponding Nyquist diagram (see Fig. 3.1), and applying the Nyquist
criterion, it can be concluded that η = 0 is an asymptotically stable equilibrium point
for system (3.35). This means that the averaged value ηa, of the real world input η(t) in
(3.34) will asymptotically approach the desired averaged signal ηa given by controller 3.27.
Hence, since the closed-loop averaged dynamics (3.28), obtained under such control action
is asymptotically stable, the modified real world control input (3.34) cannot introduce
undesired permanent oscillations.
The overall system stability under this voltage stabilizing solution can be proved by
exploiting the same two-time scale averaging theory and input to state stability arguments
used in 3.3. In particular, replacing (3.34) into (3.4), the following closed-loop current
73
Chapter 3. ON THE CONTROL OF DC-LINK VOLTAGE IN SHUNT ACTIVE FILTERS
−0.8−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2−1
−0.5
0
0.5
1
1.5
ReIm
Figure 3.1: Nyquist diagram of system (3.37).
system is obtained
˙x = (M(R,L)− 1
LK)x− 1
LΓξ + f(za, θ, ˙za,
˙θ, ǫ)
˙ξ = Φξ +Qx.
(3.38)
where
f(za, θ, ˙za,˙θ, ǫ) = Tkp ¨za + ǫki ˙za + kp ˙za(t− T/2)− ˙
θ(t− T/2)
+M(R,L)[−Tkp ˙za − ǫkiza − kpza(t− T/2) + θ(t− T/2)].(3.39)
Then, the boundary layer system can be derived taking ǫ = 0
˙x = (M(R,L)− 1
LK)x− 1
LΓξ + f(za, θ, 0, 0, 0)
˙ξ = Φξ +Qx.
(3.40)
and defining the change of variables, similar to what in (3.12)
ζa = Πξ −ΠRξ(θ(t− T/2)− kpza(t− T/2)) + LGx (3.41)
it can be verified that the same linear systems as (3.13) is obtained. Finally, as for Prop.
3.3.2, practical stability of the interconnection between the current and the averaged
voltage subsystems can be proved recalling results of Theorem 1 in [76].
74
Part II
Explicit Saturated Control Design
75
Chapter 4
Control of Linear Saturated
Systems
This chapter outlines the main analysis and synthesis results regarding direct sat-
urated feedback law design for input constrained linear systems. Among the possi-
ble strategies, the focus is put in particular on a LDI-based representation of sat-
urated linear systems, which, beside reducing conservatism with respect to stan-
dard sector characterization, naturally leads to LMI-based control design meth-
ods. Design algorithms to optimally deal with common control theory problems
are presented, then some improvements, provided by adopting various classes of
non-quadratic Lyapunov functions, possibly combined with non linear controllers,
are discussed.
4.1 Introduction
In the previous chapters, saturation nonlinearity at the control inputs has been neglected
at first control design stage, focusing on how to introduce efficient anti-windup systems to
cope with saturation adverse effects. Here systematic strategies, accounting for saturation
at the outset of the control law design, are discussed. The modern approach to address
this issue, is a Lyapunov based procedure, where some quantitative measures such as the
size of the domain of attraction, the L2 gain or the convergence rate are systematically
characterized for the saturated closed-loop system.
Two main steps are involved in this procedure; first a proper characterization of the
saturation (or deadzone) nonlinearity has to be provided, including it into a sector (as
recalled in A.5), or describing the system by means of LDIs ([21], [22]). The second step
exploits tools from absolute stability theory, or an LMI characterization of stability and
performance ( [17], [78]), respectively, to formally analyze the saturated system properties
or for control design purposes.
Here the focus is mainly put on the aforementioned LDI-based framework, since it will be
the base to cope with the power electronics application of ch.5.
76
4.2. Reducing conservatism in saturation nonlinearity characterization
In this respect the following class of saturated linear systems is considered
x = Ax+Bp+Bww, p = sat(u)
u = Cux, z = Czx(4.1)
where x ∈ Rn, u ∈ R
m, and sat(·) is a unit saturation defined as in (1.1), while w(t) is an
exogenous disturbance input and z(t) is the performance output.
In 4.2, 4.3, the main analysis and synthesis results will be presented for linear static state
feedback control laws and relying on quadratic Lyapunov candidates. However it will be
showed how the approach can be easily extended to the output feedback case. While in 4.4
the potential enhancement given by nonlinear control laws, combined with non-quadratic
Lyapunov functions is presented.
4.2 Reducing conservatism in saturation nonlinearity char-
acterization
The most popular solution to explicitly deal with input constraints, is by the sector char-
acterization reported in A.5. The payoff in adopting this mathematical description, is
that saturation nonlinearity is expressed in terms of quadratic inequalities, that, com-
bined with quadratic Lyapunov candidates, allow to describe the system properties by
means of LMI constrained problems, for which reliable and efficient solution algorithms
are available (see A.2). In order to motivate this claim, consider a quadratic Lyapunov
candidate V (x) = xTPx (P = P T > 0) for system (4.1). Assume w(t) = 0, then, applying
(A.29), it’s easy to proof that a sufficient condition to ensure asymptotic stability for all
the closed-loop trajectories is ([75]):
V (xp) = xT (ATP + PA)x+ xT (PB +BTP )p, ∀x 6= 0 : pT (p− Cux) ≤ 0 (4.2)
by applying S-procedure to the two quadratic inequalities the following condition is ob-
tained [
ATP + PA PB + CTu T
BTP + TCu −2T
]
< 0, T = diag(τ1, . . . , τm) ≥ 0. (4.3)
Condition (4.3) is an LMI in the variable P as long as Cu is given; if the synthesis problem
is concerned, it suffices to multiply the above inequality on the left and the right by
diag(P−1, T−1) to get a linear inequality in the variables Q = P−1, U = T−1, Y = QCu
[
QAT +AQ BpU + Y T
UBTp + Y −U
]
< 0 (4.4)
similar considerations can be made as concerns external stability and other common control
system properties (see [75] for details). The main drawback of this methodology is that
asymptotic stability of the plant is required, otherwise the LMI conditions (4.3), (4.4)
would be clearly unfeasible. In order to overcome this strong limitation, a possible solution
is to define local sector bounds, and apply the absolute stability tools over a finite region
77
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
of the state space. In [15], the following proposition, corresponding to a local multivariable
circle criterion, is presented;
Proposition 4.2.1 Assume that (A, Cz, B) is controllable and observable, given an el-
lipsoid E(P, ρ), if there exist positive diagonal matrices K1, K2, with K1 < I, K2−K1 ≥ I
such that
(A+BK1Cu)TP + P (A+BK1Cu) +
1
2(CT
uK2 + PB)(K2Cu +BTP ) < 0 (4.5)
and E(P, ρ) ⊆ L(K1Cu), where L(K1Cu) := x| K1Cu,ix ≤ 1, i ∈ [1,m] is the linear
region of the feedback law, then E(P, ρ) is a contractive positive invariant set, i.e the
trajectories entering it remain in it and then converge to the origin.
In [16], [21] this result is used to extend stability analysis and synthesis for unstable plants,
estimating the basin of attraction by the maximum volume ellipsoid satisfying (4.5). How-
ever, since inequality (4.5) is not jointly convex in K1, K2, P , the approaches involve
bilinear matrix inequalities, that require a larger computational burden to be solved.
An alternative approach, extensively discussed in [17], is to include the saturated system
into a polytopic model, placing the saturated control action p = sat(Cux) inside the convex
hull of a group of linear feedback laws. In this way, the system properties can be charac-
terized by more tractable LMI conditions, since the hard input nonlinearity “disappears”
in the polytopic differential inclusion. Before introducing the main results regarding this
approach, some preliminaries about convex hull properties need to be recalled ([22], [17])
Lemma 4.2.2 Given u ∈ Co ui : i ∈ [1, n1], v ∈ Co vj : j ∈ [1, n2], then[
u
v
]
∈ Co
[
ui
vj
]
(4.6)
Proof Rewrite u and v as u =∑n1
i=1 αiui, v =∑n2
j=1 βjvj , with∑n1
i=1 αi =∑n2
j=1 βj = 1.
Then [
q
v
]
=
[∑n1i=1 αiui
∑n2j=1 βjvj
]
=
[∑n1i=1 αiui(
∑n2j=1 βj)
∑n2j=1 βjvj(
∑n1i=1 αi)
]
=
n1∑
i=1
n2∑
j=1
αiβj
[
ui
vj
]
. (4.7)
Noting that∑n1
i=1
∑n2j=1 αiβj = 1, (4.6) is obtained.
Now define D as the set of m×m diagonal matrices having 1 or 0 as diagonal entries, and
denote the 2m elements of the set as Di, while Im −Di, which is still an element of D, is
denoted as D−i . Then the following fact holds
Lemma 4.2.3 Given two vectors u, v ∈ Rm, with |vi| ≤ 1 ∀ i ∈ [1,m], then
p = sat(u) ∈ CoDiu+D−
i v, i ∈ [1,m]
(4.8)
Proof By the assumption |vi| ≤ 1 if follows pi = sat(ui) ∈ Co ui, vi, ∀ i ∈ [1,m]. Thus,
applying Lemma 4.2.2 inductively, inclusion (4.8) easily follows.
78
4.2. Reducing conservatism in saturation nonlinearity characterization
Now it is trivial to verify that the same fact holds if u, v in (4.8) are respectively replaced
by two feedback laws Cux, Hux, with ‖Hux‖∞ ≤ 1, obtaining the inclusion
p = sat(Cux) ∈ CoDiCux+D−
i Hux, i ∈ [1, 2m]. (4.9)
which, as mentioned, places the saturated control law into a family of linear state feedback
regulators. With this result at hands, system (4.1) can be finally represented by the
following polytopic differential inclusion
x ∈ CoAx+B(DiCux+D−
i Hux) +Bww
z = Czx.(4.10)
It’s further to remark that the above inclusion holds in general only locally, specifi-
cally in the linear region L(Hu) := x : ‖Hux‖∞ ≤ 1 of the auxiliary matrix Hu which
parametrizes the inclusion. Therefore an additional degree of freedom can be injected in
a finite region of the state space, where the system is expected to operate, reducing con-
servatism with respect to the criterion 4.5. Moreover, description (4.10) allows to check
the (possibily local) quadratic stability properties of the original nonlinear system by a
simple LMI condition, as stated by the following result ([22])
Theorem 4.2.4 Consider the closed loop system (4.1) with w = 0, given an ellipsoid
E(P, ρ), if there exist a matrix Hu ∈ Rm×n such that
(A+Bp(DiCu +D−i Hu))
TP + P (A+B(DiCu +D−i Hu)) < 0, ∀ i ∈ [1, 2m] (4.11)
and E(P, ρ) ⊆ L(Hu), i.e Hu,ix ≤ 1, ∀ x : xTPx ≤ ρ, i = 1, . . . ,m, then E(P, ρ) is a
contractive invariant set.
Proof Let V (x) = xTPx, invariance condition of E(P, ρ) can be expressed as
V = 2xT (Ax+Bpsat(Cux)) < 0, ∀ x ∈ E(P, ρ). (4.12)
As Hu,ix ≤ 1 ∀ x ∈ E(P, ρ), by (4.9) it follows
Ax+Bsat(Cux) ∈ coAx+B(DiCu +D−i Hu)x, , i in [1, 2m] (4.13)
and then
V ≤ maxi∈[1,2m]
2xTP (A+Bu(DiCu +D−i Hu))x ∀ x ∈ E(P, ρ). (4.14)
Since (4.11) holds by hypothesis, it turns out that
maxi∈[1,2m]
2xTP (A+B(DiCu +D−i Hu))x < 0∀ x 6= 0
which verifies (4.12).
By simple computations, it can be showed how restricting Hu = K1Cu, condition (4.11)
is equivalent to the so called vertex criterion ([15] Th. 10.4), which is less conservative
79
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
then the local sector condition (4.5), but, obviously more restrictive then (4.11), where
Hu can be arbitrary selected. Alternatively, note that the ellipsoid E(P, ρ) can go beyond
the linear region of the saturation function sat(Cux), as long as an auxiliary feedback law
containing it in its linear region can be found. Furthermore the auxiliary law does not
need to be linear, indeed by lemma 4.2.3, it’s easy to prove theorem 4.2.4 for any h(x) such
that ‖h(x)‖∞ ≤ 1. In 4.4 nonlinear laws, together with non quadratic Lyapunov functions,
will be exploited, to improve the results presented in the following section. Finally, results
in 4.2.4 can be tightened for single input systems, obtaining a necessary and sufficient
quadratic invariance condition (see [17] ch. 7-8 for details).
4.3 Saturated control design via LMI constrained optimiza-
tion techniques
Based on the polytopic inclusion (4.10), and quadratic Lyapunov candidates, several con-
trol theory problems can be solved for the original saturated system, by deriving conditions
in the form of LMIs. From these conditions, convex problems can be formulated in order
to obtain an optimal estimation of the system properties or, if synthesis problems are
concerned, select the optimal feedback law which induces the desired saturated system
behavior. In the next Subsections the most common issues are discussed.
4.3.1 Domain of attraction maximization
Invariance condition (4.11), can be exploited to establish the stability properties of (4.1),
in particular, the system region of attraction can be approximated with the “largest”
contractive invariant ellipsoid satisfying (4.11). In this respect, a natural choice is to
maximize the convex quantity logdet(P−1), that is directly related to the ellipsoid volume
(see [75]), under the LMI constraint (4.11). However, in principle a high volume set
could be overly stretched on some directions and very thin along others. This situation
would clearly affect the ensured stability margin. Hence it can be profitable to optimize
the ellipsoid dimensions with respect to a specific shape reference set XR, so that the
invariant region can be ensured to have a certain size along the desired directions, and
informations on the initial conditions can be exploited. To this aim, the objective function
can be modified to maximize a scalar α with the additional constraint E(P ) ⊆ αXR, where
αXR := αx : x ∈ XR. In conclusion, the maximal (w.r.t XR) quadratic stability region
for (4.1)can be obtained selecting the matrices P ∈ Rn×n, Cu, Hu ∈ R
n×n as the optimal
variables of
supP>0,ρ,Hu,Cu
α
s.t. αXR ⊂ E(P, ρ)E(P, ρ) ⊂ L(Hq)
(A+Bp(DiCu +D−i Hu))
TP + P (A+Bp(DiCu +D−i Hu)) < 0, ∀ i ∈ [1, 2m].
(4.15)
80
4.3. Saturated control design via LMI constrained optimization techniques
The problem constraints can be cast into LMIs, under some hypothesis. Assume XR is a
convex set, then the first inclusion can be rewritten as a matrix inequality, e.g. if XR is
an ellipsoidxTRx ≤ 1
, the inclusion is equivalent to α2 P
ρ − R > 0, which by Schur’s
complement yields
(
Pρ
)−1
I
I γR
≥ 0, γ = 1/α2. (4.16)
Similar reasoning can be made for the set inclusion E(P, ρ) ⊂ L(Hq), which can be verified
to be equivalent to minxTPx : |Hu,i| = 1, i ∈ [1,m]
≥ ρ. The minimum can be com-
puted by Lagrange multipliers obtaining (Hq,iP−1HT
q,i)−1, which yields ρHq,iP
−1HTq,i ≤
1, i = 1, . . . , 1 and, by Schur’s complement
1 Hu,i
(
Pρ
)−1
(
Pρ
)−1
HTu,i
(
Pρ
)−1
≥ 0, i ∈ [1,m]. (4.17)
Note that if ρ → ∞, inequality (4.17) enforces Hu → 0, and global results are recovered.
Now let Q =
(
Pρ
)−1
, Z = HuQ, Y = CuQ; by multiplying the third inequality in (4.15)
on left and right by Q, and rewriting (4.16), (4.17) in the new variables, the following
EVP is obtained
infQ>0,Z
γ
[
1 Zi
ZTi Q
]
≥ 0, i = 1, . . . ,m
[
Q I
I γR
]
≥ 0
QAT +AQ+ (DiY +D−i Z)
TBT +B(DiY Q+D−i Z) < 0, i ∈ [1, 2m]
(4.18)
thus the optimal feedback gain matrix can be recovered as C∗u = Y ∗Q∗−1, where Y ∗, Q∗−1
are the optimal values of (4.18).
If the auxiliary feedback matrix Hu is restricted to be equal to Cq, i.e Y = Z, the set of
2m inequalities in (4.18) reduces to the inequality QAT +AQ+ZTBTp +BpZ < 0. Thus, a
reduced optimization problem is obtained, whose optimal solution cannot be better than
what obtained for (4.18), since the degrees of freedom have been reduced. On the other
hand, since 2m−1 constraints have been eliminated, the minimum of the reduced problem
cannot be larger than the one in (4.18). From that arguments, it can be concluded that,
if the only purpose is to enlarge the domain of attraction, a simpler problem with Y = Z
can be considered. As we will be showed in the following, the freedom in choosing Hu (Z)
can be exploited to meet other specifications, beyond the domain of attraction.
81
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
The same procedure is still valid if a dynamic output feedback control in the form
x = Ax+Bup, p = sat(u)
ξ = Acξ +Bcz
z = Czx, u = Ccξ +Dcz
(4.19)
is concerned. Defining the state vector x = [x ξ]T , the closed loop state-space equation
can be written as
˙x = Ax+ Bsat(Kx)
A =
[
A 0
BcCz Ac
]
, B =
[
B
0
]
, K = [DcCz Cc](4.20)
which is in the same form as (4.1).
A polytopic description similar to (4.10) can be derived also to characterize the deadzone
nonlinearity ([23]), and more general memoryless nonlinearities ([20]).
4.3.2 Disturbance rejection with guaranteed stability region
Now we move to analyze external stability of system (4.1), in particular L2 disturbances
w(t) are considerd. As mentioned in ch. 1, since in principle a saturated system may not
have a well defined (finite) L2 gain for any disturbance energy levels, and the gain can in
general depend on the value of ‖w(t)‖2, it would be profitable to characterize the system
rejection property via a nonlinear L2 function. The parametrized PLDI (4.10) is suitable
to carry out the nonlinear L2 gain analysis.
The first step is to solve the so called disturbance tolerance problem ([19]), i.e to determine
the maximum energy level smax, such that for any ‖w‖2 ≤ smax, the trajectories of the
closed-loop system (4.1) are bounded. Two different situations need to be distinguished;
zero and nonzero initial condition. Starting with the assumption x(0) = 0, relying on
description (4.10) and a quadratic Lyapunov candidate, the the problem can be approached
by establishing a sufficient condition under which the trajectories starting from the origin,
and perturbed ‖w‖2 ≤ s < smax, are kept inside an outer ellipsoid. In this respect results
have been established in ([19])
Theorem 4.3.1 Consider system (4.1) under a given feedback law u = Cux and let also
P > 0 be given. If there exist a matrix Hu ∈ Rm×n and a positive scalar η, such that
(A+B(DiCu +D−i Hu))
TP + P (A+B(DiCu +D−i Hu)) +
1
ηPBwB
TwP < 0, ∀ i ∈ [1, 2m]
E(P, sη) ⊂ L(Hu)
(4.21)
then the trajectories of the closed loop system starting from the origin will remain inside
E(P, sη) for all w s.t. ‖w‖22 ≤ s
Proof Consider a quadratic Lyapunov function V (x) = xTPx, the derivative of V along
the closed-loop system trajectories is V = 2xTP (Ax + Bsat(Cux) + Bww. Now let the
82
4.3. Saturated control design via LMI constrained optimization techniques
ellipsoid E(P, ρ) and matrix Hu such that E(P, ρ) ⊂ L(Hu); following the same procedure
as the proof of Th. 4.2.4 it follows
V (x) = 2xTP (Ax+Bsat(Cux) ≤ maxi∈[1,2m]
2xTP (A+B(DiCu+D−i Hu))x, ∀ x ∈ E(P, ρ)
(4.22)
and, by Young’s inequality
2xTPBww ≤ 1
ηxTPBwB
TwPx+ ηwTw ∀η > 0 (4.23)
thus V (x,w) can be upper bounded inside E(P, ρ) as
V (x,w) ≤ maxi∈[1,2m]
2xTP (A+Bp(DiCq +D−i Hq))x+
1
ηxTPBwB
TwPx+ ηwTw. (4.24)
Now set ρ = sη, by (4.21) and integrating both sides of(4.24), it follows
V (x(t)) ≤ η
∫ t
0w(τ)Tw(τ)dτ ≤ sη (4.25)
which proves the theorem.
With this result at hand, the maximum tolerable disturbance energy level smax can be
estimated by solving the problem
supP>0,Hu
s
s.t. (A+Bp(DiCu +D−i Hu))
TP + P (A+Bu(DiCu +D−i Hu)) +
1
ηPBwB
TwP < 0, ∀ i ∈ [1, 2m]
E(P, s) ⊂ L(Hu)
(4.26)
then, assuming without loss of generality η = 1, performing the change of variables Q =
P−1, ν = 1/s, Z = HuQ, and expressing the set inclusion by means of Schur’s complement,
(4.26) is cast into the LMI constrained problem
infQ>0,Z,Y,ν
ν
s.t. QAT +AQ+ (BDiCuQ)T + (BDiCuQ) + (BD−i Z)
T + (BD−i Z) +BwB
Tw < 0, i ∈ [1, 2m]
[
ν Zi
ZTi Q
]
≥ 0, i = 1, . . . ,m.
(4.27)
The next step is to compute the L2 gain of the system restricting the analysis to exogenous
inputs satisfying ‖w‖2 ≤ smax. In this respect the following result can be established ([17],
[19])
Theorem 4.3.2 Let smax be the maximal tolerable disturbance level determined by (4.27).
Given an arbitrary γ > 0, if there exist a matrix Hu such that
(A+B(DiCu +D−i Hu))
TP + P (A+Bu(DiCu +D−i Hu)) + PBwB
TwP +
1
γ2CTC ≤ 0, i ∈ [1, 2m]
E(P, s) ⊂ L(Hu)
(4.28)
then the L2 gain of system (4.1) from w to z is less then γ for any ‖w‖22 ≤ smax.
83
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
Proof Consider a quadratic Lyapunov function V = xTPx, by (4.9), the derivative along
system trajectories can be expressed as
V (x,w) ≤ maxi∈[1,2m]
2xTP (A+B(DiCu +D−i Hu))x+ xTPBwB
TwPw+wTw ∀ x ∈ E(P, s)
(4.29)
combining this inequality with (4.28) yields
V (x,w) ≤ wTw − 1
γ2xTCTCx = wTw − 1
γ2zTCT z (4.30)
which is the standard external stability condition already exploited in (1.15).
As a product of this result, the system disturbance rejection level can be evaluated by
computing the tightest upper bound of system restricted L2 gain, solving the problem
infP>0,Hu,Cu
γ2
s.t. E(P, s) ⊂ L(Hu)
(A+Bp(DiCq +D−i Hu))
TP + P (A+Bp(DiCq +D−i Hu)) PBw CT
BTwP −I 0
C 0 −γ2I
≤ 0
(4.31)
that can be cast into a convex problem by performing the previously mentioned tranfor-
mations. Finally, the nonlinear L2 gain function can be computed by solving (4.31) for s
ranging over [0, smax].
Now we move to consider the nonzero initial state situation, since for nonlinear systems
the effect of the initial condition may not vanish as time goes on, a possible way to measure
the rejection capability is to compare the relative size of two nested sets ([19]); one in-
cluding the set of initial conditions, and the other eventually bounding all the trajectories
originating from the former. In this respect, the following extension of Th. 4.3.1 can be
stated
Theorem 4.3.3 Consider system (4.1) under a given feedback law u = Cux and let P > 0
be given. If there exist an Hu and a positive scalar η, such that
(A+B(DiCu +D−i Hu))
TP + P (A+B(DiCu +D−i Hu)) +
1
ηPBwB
TwP < 0, i ∈ [1, 2m]
E(P, 1 + sη) ⊂ L(Hu)
(4.32)
then the trajectories of the closed loop system starting from E(P, 1) will remain inside
E(P, 1 + sη) for all w s.t. ‖w‖22 ≤ s.
the proof can be carried put quite similarly to that of Th. 4.3.1 (see [19] for the details).
As for the previous case, condition (4.32) can be exploited to approximate the largest
disturbance the closed-loop system can tolerate, provided that the trajectories start inside
a given ellipsoid E(S, 1). In other words, the largest s, such that the nested ellipsoids
84
4.3. Saturated control design via LMI constrained optimization techniques
defined in Th. 4.3.3 exist, and E(S, 1) ⊂ E(P, 1), is sought for. Formally the following
problem is formulated
supP>0,η,s
s
s.t. (A+B(DiCu +D−i Hq))
TP + P (A+B(DiCu +D−i Hu)) +
1
ηPBwB
TwP < 0, i ∈ [1, 2m]
(1 + sη)Hu,iP−1HT
u,i ≤ 1, i ∈ [1,m][
S I
I P−1
]
> 0
(4.33)
where the set inclusion constraints have been already transformed into matrix inequalities.
Defining µ = 1/(1 + sη), Q = P−1, Z = HqQ an EVP is obtained for fixed µ. Then the
global optimum can be in principle derived sweeping µ over [0, 1].
Bearing in mind these considerations, η can be associated to the disturbance rejection
level of the system, since it represents an index of the size difference between the ellipsoid
E(S, 1), which is ensured to contain the initial state, and the invariant E(P, 1 + sη). For-
mally speaking, similarly to what reported in 2.6 for SAFs saturated control, given a set of
initial conditions E(S, 1) and the maximal disturbance energy smax the smallest invariant
ellipsoid containing E(S, 1) can be computed as
infP,Hu
η
E(S, 1) ⊂ E(P, 1)
(A+Bp(DiCu +D−i Hu))
TP + P (A+B(DiCu +D−i Hu)) +
1
ηPBwB
TwP < 0, i ∈ [1, 2m]
E(P, 1 + sη) ⊂ L(Hu).
(4.34)
It’s further to remark that the differential inclusion representation can be exploited, in
a similar fashion, to analyze the saturated system rejection properties for other common
classes of exogenous inputs, such as norm bounded persistent disturbances ([22]) or peri-
odic signals ([79]).
4.3.3 Convergence rate maximization
In 4.3.1 the focus was on ensuring a large stability region for saturated input systems, how-
ever, beside stability, another common requirement is to ensure a fast system response,
namely a high system convergence rate. Solving problem (4.18) can lead to a closed-loop
state matrix A + BCu whose eigenvalues are very close to the imaginary axis, thus pro-
ducing a sluggish response. On the other hand, in order to ensure a fast response, an high
gain feedback matrix Cu is usually required, which, due to saturation nonlinearity, is in
contrast with a large basin of attraction specification. Here feedback design techniques to
maximize the convergence rate of saturated systems in the form (4.1) are recalled, then
a method to obtain a trade-off between the two contrasting objectives regarding stability
85
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
and responsiveness of the system is defined.
Before elaborating on the convergence rate maximization, it’s further to define for con-
venience the convergence rate of system (4.1) on a Lyapunov function level set LV (ρ)
as
α =1
2min
− V (x)
V (x)
∀x ∈ LV (ρ). (4.35)
In particular, as in the previous Subsections, quadratic functions having ellipsoidal level
sets will be considered. Furthermore a disturbance free case w(t) = 0 will be considered
for the sake of simplicity.
The problem of maximizing the overall convergence rate of systems in the form (4.1) has
a well known solution in the optimal time bang-bang law minimizing V (i.e maximizing
the decay rate) ui = −sign(BTi Px), i ∈ [1,m]. However, discontinuity of such feedback
law can give rise to chattering phenomena for practical implementation, moreover the
resulting closed-loop dynamics should be carefully analyzed to exclude finite escape time
of the system trajectories for some initial conditions. A simple solution to overcome this
drawbacks is to replace the bang-bang controller with a saturated high gain feedback law,
at the cost of some optimality. The reduction in the convergence rate, and the stability
property of the modified controller can be formally characterized as follows
Theorem 4.3.4 Assume that an ellipsoid E(P, ρ) can be made contractive invariant with
a bounded control law, then there exists a k0 > 0 such that, for any k > k0, E(P, ρ) is a
contractive controlled invariant set under the feedback law
u = −sat(kBTPx). (4.36)
Proof The assumption that E(P, ρ) can be made contractive invariant by a bounded law
implies it can be made invariant also by the bang-bang control (4.36), that is
V = xT (ATP + PA)x− 2
m∑
i=1
xTPBisign(BiPx) < 0 ∀x ∈ E(P, ρ) (4.37)
to prove the theorem it has to be shown that this is equivalent to
V = xT (ATP + PA)x− 2m∑
i=1
xTPBisat(kBiPx) < 0 ∀x ∈ E(P, ρ). (4.38)
Since V is an homogeneous function, condition (4.38) can be equivalently checked on the
boundary of ∂E(P, ρ). Define
ǫ = − maxx∈∂E(P,ρ)
xT (ATP + PA)x− 2m∑
i=1
xTPBisign(BiPx)
(4.39)
which by (4.37)is positive. After some computations it follows
xT (ATP+PA)x−2m∑
i=1
xTPBisat(BiPx) ≤ −ǫ+2m∑
i=1
xTPBi(sign(BTi Px)−sat(kBT
i Px))
(4.40)
86
4.3. Saturated control design via LMI constrained optimization techniques
As
∣∣xTPBi(sign(B
Ti Px)− sat(kBT
i Px))∣∣ =
0 if |kBTi Px| > 1
≤ 1k if |kBT
i Px| ≤ 1(4.41)
the absolute value of the sum in (4.40) can be upper bounded with 2mk , hence, choosing
k > k0 =2mǫ , by (4.40) it turns out
xT (ATP + PA)x− 2m∑
i=1
xTPBisat(kBiPx) < 0 ∀x ∈ ∂E(P, ρ) (4.42)
which proves (4.38).
This result states that, albeit under law (4.36) the convergence rate is slightly reduced, the
same invariant set of the bang-bang controller can be obtained. It is worth noticing that an
high gain is not needed to ensure invariance, but only to provide a fast convergence rate,
since if the ellipsoid can be made invariant with a bounded control (formal procedures to
check this can be found in [17] ch. 11), there exists a k > 0 such that E(P, ρ) is invariantunder (4.36).
The next step is to investigate how the decay rate and the size of the invariant ellipsoid
are related, i.e to determine how α depends on ρ, and P . As regards the dependence on
ρ the following results can be proved ([17] Th. 11.2.4)
• α(ρ) = 12min
− Vρ : xTPx = ρ
• There exists a limit value
β∗ = min−xT (ATP + PA)x : xTPx = 1, xTPB = 0
(4.43)
such that limρ→0 α(ρ) =β∗
2 .
As expected α increases as ρ is decreased, i.e the size of the invariant ellipsoid is shrunk
in face of a higher convergence rate. Furthermore α approaches a finite limit as ρ tends
to zero, hence β∗ is an index on the maximum convergence performance, inside E(P, ρ),of system (4.1). If the matrix P is given, β∗ is derived by definition (4.43) as β∗ =
−λmax((NTPN)−1NT (ATP+PA)N), where N is a basis for the Kernel of BTP . However
it would be profitable to select the matrix P so that the resulting β∗, and then the
convergence rate, is not too small. For this purpose the following proposition (see [17] ch.
11) can be exploited to derive an LMI-based procedure to shape P
Proposition 4.3.5 Let P be given, then
β∗ = supCu
η
s.t. (A+BCu)TP + P (A+BCu) ≤ −ηP.
(4.44)
Thus, in principle, letting P to be a free parameter, and assuming (A,B) in (4.1) is a
controllable pair, problem (4.44) can be solved to make −β∗/2 equal to the largest real part
87
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
of the eigenvalues of the matrix A+BpCq, according to the decay rate definition of linear
systems. Thus, in principle β∗ can be made arbitrary large, increasing the convergence
rate until specification on system response are met. On the other hand, in general, as β∗
increases, matrix P becomes badly conditioned, affecting the shape of the ellipsoidal level
set E(P, ρ), that can result very small with respect to a fixed shape reference set. Roughly
speaking the invariant ellipsoid could be very “thin” along some state space directions,
reducing the stability margin.
A possible solution to found a suitable balance between a large domain of attraction and
a fast convergence rate consist in a suitable combination of the two problems (4.18), and
(4.44). For example, if a lower bound β is specified for β∗, then the control law maximizing
the size of the domain of attraction, and providing at the same time a convergence rate
greater then β, can be derived by solving the following mixed problem
supP>0,ρ,Cu,Hu
α
s.t. αXR ⊂ E(P, ρ)E(P, ρ) ⊂ L(Cu)
(A+BpCu)TP + P (A+BpCu) < 0
(A+BpHu)TP + P (A+BpHu) ≤ −βP
(4.45)
which is exactly problem (4.18), except for the last constraint inequality, that has been
added to guarantee a minimal convergence rate, related to β. Following the same reasoning
reported in 4.3.1, it is easy to verify that (4.45) can be cast into a convex LMI-constrained
problem. The above analysis can be further extended, with suitable modifications, to the
case of system perturbed by peak bounded disturbances (see [17] for details).
4.4 Improvements via non-quadratic Lyapunov functions
The effectiveness of the LDI appraoche considered in this chapter depends on two factors:
how well the LDI description approximate the original saturated system, and what tools
are exploited to analyze it and synthesize control laws.
As regards this second factor, the results presented so far are based on quadratic Lya-
punov functions that, as mentioned, are a natural choice due to the vast amount of tools
available for their use, and the possibility to convert stability and performance problems
of LDIs, into LMI constrained optimization problems. On the other hand, it’s well known
that quadratic forms are not a universal class for system described by LDIs, therefore,
as showed in the previous section, only sufficient conditions can be derived. In other
words, there are cases where an LDI is stable but a quadratic Lyapunov function does not
exist. Furthermore, even when a quadratic Lyapunov can be found, it usually provides
conservative results, expecially as concerns regional stability and performance analysis for
constrained systems.
In this section, some classes of non-quadratic Lyapunov candidates, stemming from robust
88
4.4. Improvements via non-quadratic Lyapunov functions
control of time-varying and uncertain systems, which, in the considered framework, share
the polytopic representation with input saturated systems, are presented.
In the following the autonomous PLDI
x ∈ co Aix (4.46)
with Ai = (A + BDiCu + BD−i Hu) will be considered to represent system (4.10) in case
the auxiliary feedback matrix Hu has already been defined (e.g by means of the quadratic
stability tools established in the previous sections), while the controlled inclusion
x ∈ co Aix+Biv (4.47)
where Ai = (A + BDiCu), Bi = BD−i , v = Hux will be exploited to represent the case
when Hu is synthesized by means of a non quadratic control Lyapunov candidate. It’s
worth to recall that LDI (4.10) holds only locally, inside a particular level set of the con-
sidered Lyapunov function that, in case of quadratic forms, has an ellipsoidal form. Here,
without loss of generality, the unit invariant LV (1) := x : V (x) ≤ 1 will be considered
to replace the condition used in 4.2, with the inclusion LV (1) ⊂ L(Hu). It will be showed
how less restrictive conditions can be obtained, for both analysis and synthesis problems,
finally, since the potential of non-quadratic Lyapunov functions is fully unleashed only if
associated with nonlinear control laws ([80]), an example on how to combine this two tools
to further reduce conservatism will be sketched for a particular class of functions, which
has been recently proved to be universal for LDIs ([26]).
As mentioned, the price for these improvements is an increased complexity in the opti-
mization problems to be solved for analysis and feddback design purposes. In general
non-convex BLMI constrained problems need to be solved. Although some effective algo-
rithms have been proposed to deal with issues ([81], [82]), BLMI problems are not yet a
mature technology, in the sense that they cannot be straightforwardly solved by running
reliable algorithm, without having a deep knowledge of the mathematical details. Further-
more they require a higher computational burden with respect to LMI problems, but given
the fast growth of nowadays processors technology, this could not be a strong limitation.
4.4.1 Piecewise quadratic Lyapunov functions
A natural choice to extend the quadratic invariance conditions given in 4.2, is to adopt
piecewise quadratic functions. It’s easy to guess that, searching for a common quadratic
Lyapunov function for all the member systems of the polytopic inclusion can be very lim-
iting. Indeed this has been established in the literature by long time, as in the context of
absolute stability, the Lure Postinikov functions ([15] ch. 7), associated with the Popov
criterion in the frequency domain, which is known to be less conservative than the classic
circle criterion (springing from quadratic stability considerations), can be seen as a par-
ticular case of piecewise quadratic functions.
In [34], [83] a more general piecewise quadratic representation, based on a suitable state
89
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
space partition, has been proposed for the following class of piecewise affine systems
x = Ai + ai, x(t) ∈ Xi (4.48)
where Xi are suitably defined cells covering a finite region or the entire state space. The
analysis is presented for this class of systems, since it will be exploited in chapter 5
to refine the stability analysis of a class of bilinear saturated systems arising in power
electronic applications. However it is easy to adapt all the results here presented, for
systems in the form (4.46) or (4.47). Consider a Lyapunov function composed by different
quadratic “pieces” which stabilize the system whose dynamics is defined by Ai inside the
cell Xi where this representation holds. The basic idea is to define a quadratic Lyapunov
function for each cell Xi, in this way conservatism is clearly reduced, since each “piece”
has to provide stability only for the dynamics of the cell it is associated with.
For the sake of simplicity, assume the cells Xi are closed polyhedral sets with disjoint
interiors, and that ai = 0 the for the cells containing the origin. Now let the Lyapunov
candidate function be V (x) = xTPix, for x ∈ Xi; in order to ensure continuity across the
boundaries of the partitioning cells the following conditions need to be fulfilled
Pi = F Ti TFi
Fi
[
x
1
]
= Fi
[
x
1
]
if x ∈ Xj ∩Xi
Fi = [Fi fi].
(4.49)
The existence of parametrizing matrices Fi, ensuring continuity, is guaranteed by the
hypothesis of polyhedral cells, furthermore a solution of the above conditions satisfying
fi = 0 for the sectors containing the origin can always be computed. Note that matrices
Pi depends linearly on the decision variables that are collected in the symmetric matrix
T . Hence the search for a piecewise quadratic Lyapunov function for autonomous LDIs as
(4.48), is still an LMI problem.
Now denote with I0 the set of indexes i such that Xi contain the origin, the candidate
Lyapunov function can be written in the general form
V (x) =
xTPix if x ∈ Xi, i ∈ I0
x
1
T
Pi
x
1
if x ∈ Xi, i /∈ I0(4.50)
where Pi = FiTFi, Pi = FiT Fi. The main result regarding this representation can thus
be stated ([34])
Theorem 4.4.1 Consider symmetric matrices T , Ui, Wi such that Ui, Wi have non neg-
ative entries, while Pi, Pi satisfy
ATi Pi + PiAi + ET
i UiEi < 0
Pi − ETi WiEi > 0
i ∈ I0 (4.51)
90
4.4. Improvements via non-quadratic Lyapunov functions
ATi Pi + PiAi + ET
i UiEi < 0
Pi − ETi WiEj > 0
i /∈ I0 (4.52)
where
Ei = [Ei ei], Ei
[
x
1
]
≥ 0, x ∈ Xi (4.53)
then any continuous piecewise C1 trajectory of system (4.48) converges to the origin expo-
nentially.
Proof By the definition of Ei in (4.53), it can be verified that each polyhedral cell Xi can
be characterized by the following inequalities
xTETi UiEix ≥ 0 ifi ∈ I0
x
1
T
ETi UiEi
x
1
ifi /∈ I0(4.54)
with a suitable choice of matrices Ui. Then, applying the S-procedure to the above in-
equalities and those related to positiveness of the quadratics xTPix and negativeness of
its time derivative inside the respective cells Xi, yields conditions (4.51), (4.52).
In view of this statement, a technique similar to what showed in (4.3.1) can be exploited
to estimate the system basin of attraction, searching the maximum unit level set LV (1)
contained in the region covered by the state-space partition. As in 4.3.1, a shape reference
setXR can be defined, and the scaling variable α, such that αXR ⊂ LV (1) under conditions
(4.51) can be maximized in the matrix variable T . If XR is an ellipsoid, than the previous
inclusion reads as α2V (x) ≤ xTRx ∀ x ∈ ∂XR. This condition has to be checked for each
cell, and, as it will be showed in 5.4, by S-procedure it can be formulated in terms of LMIs,
thus obtaining a generalized eigenvalue problem (GEVP).
As regards the control synthesis problem, all the previous analysis can be adopted also for
system (4.47), replacing Ai with (Ai + BiK) as far as a linear feedback control v = Kx
is concerned, or (Ai + BiKi), as far as a piecewise linear controller is concerned. In this
case the above reported matrix inequalities will become bilinear, due to the presence of
the variables K, Ki.
4.4.2 Polyhedral Lyapunov functions
In robust control theory literature ([84]) it has been well established that a class of uni-
versal Lyapunov candidates for LDIs are the so-called symmetrical polyhedral functions or
Minkowski functionals. Universality of this function class holds also when synthesis prob-
lems are concerned, i.e they can be used as control Lyapunov candidates as well. Here
the main results regarding polyhedral functions for polytopic LDIs are briefly sketched,
referring to the specific literature ([35], [85], [86]) for a complete discussion.
A polyhedral function can be expressed in the general form Vp(x) = ‖Fx‖∞, where F
91
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
is a full rank column matrix. In order to analyze the properties of system (4.46), it is
convenient to express the function in a more explicit form as
Vp(x) = maxi
|Fix| (4.55)
that is positive definite if and only if the unit ball, characterized by the polyhedral set
P := x| |Fix| ≤ 1, , contains the origin in its interior. As mentioned, this class of
functions is universal for LDIs, specifically the following necessary and sufficient condition
can be stated: a positive definite polyhedral function in the form (4.55) is a Lyapunov
function for system (4.46) if and only if there exist M matrices (i.e Mij ≥ 0, for i 6= j) Hi
such thatFAi = HiF
Hi1 ≤ −β1(4.56)
for some β > 0.
This condition can be exploited to obtain an estimation of the domain of attraction for
systems (4.1) under a given saturated feedback law. In this respect, a problem similar to
(4.15) can be formulated, by replacing the set inclusions with polyhedral sets representing
the level surfaces of function Vp(x), and the quadratic stability inequality with conditions
derived by (4.56) (see [13] ch. 4). This approach allows to eliminate conservatism intro-
duced by quadratic functions, however, by (4.56) it’s trivial to note that if the shape of
the polyhedral sets, and thus the function Vp(x) are not a priori fixed, BMI problems have
to be solved. Anyway, the solution of the arising non convex problem, is facilitated, since
polyhedral functions, as quadratic forms, are endowed with a duality property. The dual
representation of a polyhedral function is
Vp(x) = min 1α | Xα = x (4.57)
where X is the matrix whose columns are the vertices of the polyhedral function unit ball.
Hence, alternatively to (4.56), the following dual inequalities, in the M matrices variables
Li, can be checked to elaborate on stability of (4.46)
AiX = XLi
1TLi ≤ −β1T .(4.58)
As concerns the feedback synthesis problem, as mentioned the polyhedral functions have
been proved to be universal even as control Lyapunov candidates for systems in the form
(4.47). The necessary and sufficient condition associated with stabilizability of such sys-
tems is the following: the polyhedral function (4.55) is a control Lyapunov function for
system (4.47), if and only if there exist a matrix U and M matrices Pi such that
AiX +BiU = XPi
1TPi ≤ −β1T(4.59)
hold for some positive β. Beside the bilinearity of conditions (4.56), (4.59), the main
drawback of polyhedral functions is that their computation is usually not trivial, and
92
4.4. Improvements via non-quadratic Lyapunov functions
the computational burden in their construction and the solution of the above stated in-
equalities dramatically increases with the dimension of the system and the the number
of vertices of the polytope of matrices describing the LDI. This could be a limitation for
saturated systems with multiple inputs, as, by (4.10), it is easy to verify that the set of
vertices needed to define the LDI associated to the system increases exponentially with
the number of inputs m.
In order to motivate this claim, we sketch a standard iterative procedure, commonly used
to compute F ([84]), which is based on the so-called discrete Euler approximating system
(EAS) of (4.47), i.e
x(k + 1) ∈ co [I + TsAi]x+ TsBiu (4.60)
where Ts is the sampling time, and the equivalent of condition (4.59) for discrete time
systems
AkX +BkU = XPk
1TPk ≤ λ1T(4.61)
for some λ < 1.
Assuming the existence of a polytope P0 including the origin, that for convenience is
represented in the form P =x|F (0)x ≤ g(0)
, and fixing a contractive parameter λ < 1
with some tolerance, i.e λ(1 + ǫ) < 1 for a given ǫ > 0, the main steps of the procedure
can be outlined as follows;
• Set i = 0, P(0) = P;
• Form the polytope Sk :=(x, u)| F (k)([I + TsAi]x+ TsBiu) ≤ λgk
in the extended
state (x, u);
• Compute the projection of the polyhedron Sk on the subspace associated with the
state component
P(i+1) =
x| ∃u | (x, u) ∈ S(i+1)
; (4.62)
• set P(i+1) = P(i+1) ∩ P;
• if P(i) ⊆ P(i+1)(1 + ǫ) stop, else iterate the procedure.
The algorithm ensures equation (4.61) to be satisfied with λ = λ(1 + ǫ) and X the set
of vertices of the polyhedron computed as the limit of the converging sequence Pk. It
can be proved that the polyhedral function thus obtained is indeed a Lyapunov function
for the original time-continuous system, furthermore the obtained results can be exploited
to construct a linear variable structure control law for (4.47). On the other hand, it’s
easy to see that the computational burden of the above procedure clearly depends on
the system dimension and the number of vertices corresponding to the LDI description.
Moreover the obtained control laws, although continuous, are not given in an explicit form,
introducing difficulties in the implementation. In order to overcome this drawbacks, the
so-called homogeneous polynomial functions, have been proposed, allowing in particular
93
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
an explicit formulation of the control law without impairing universality, while also the
computational burden is slightly reduced. Here this class of functions is not reported for
the sake of brevity, the details can be found in [35], [87].
4.4.3 Composite quadratic Lyapunov functions
In this last Subsection a pair of so-called composite quadratic functions ([36]) is considered,
the name stems from the fact that they are composed by families of quadratic functions.
Specifically, the so-called max-quadratic functions ([23], formed by the point-wise maxi-
mum of quadratics defined by l positive definite matrices Qj
Vmax =1
2max
j=1,...,lxTQjx (4.63)
and the convex hull quadratic functions
Vc(x) =1
2minλ∈Γ
xT
l∑
j=1
λjQj
−1
x (4.64)
where Γl :=
λ :∑l
i=0 λi = 1, λi ≥ 0
will be discussed. The properties of this pair of
functions have been deeply analyzed in the literature, showing how less involved conditions
about stability and stabilizability of polytopic LDIs can be obtained with respect to stan-
dard polyhedral functions ([88], [89], [90]). Before sketching the main results concerning
this function families, some preliminary properties are recalled.
Functions (4.63), (4.64) are discussed together since they are related by a conjugacy prop-
erty in the sense of convex analysis; according to the standard definition ([91]) the conju-
gate of a convex function f(x) is given by f∗(ξ) = supx ξx− f(x). Then applying the
definition to (4.63), after some computation ([89]) it turns out that
V ∗max(ξ) = Vc(ξ) =
1
2minλ∈Γl
ξT
l∑
j=1
λjQj
−1
ξ. (4.65)
This relationship is of crucial importance, since, on the basis of convex analysis results,
it allows to develop a duality theory [88] for linear differential inclusions. In plain words,
stability of system (4.46) can be checked either associating a convex Lyapunov function
directly to it, or considering its dual dynamics
ξ ∈ coAT
i ξ
(4.66)
associated with the conjugate of the original convex Lyapunov candidates. Formally the
following results has been established ([88])
Theorem 4.4.2 Given a convex positively homogeneous of degree 2 function function
V : Rn → R, then its conjugate V ∗is convex positive definite, positively homogenenous of
degree 2, and
94
4.4. Improvements via non-quadratic Lyapunov functions
∂V (x)Ax ≤ −γV (x) ∀ x ∈ Rn, A ∈ co Ai (4.67)
∂V ∗(ξ)AT ξ ≤ −γV ∗(ξ) ∀ ξ ∈ Rn, AT ∈ co
AT
i
(4.68)
are equivalent.
This theorem, combined with the results reported in ([85]), states the equivalence of
exponential stability at the origin of systems (4.46) and (4.66). Vmax and Vp satisfies
the requirements of Th. 4.4.2: as regards Vmax it is convex (by definition), positive
homogeneous of degree 2 (i.e Vmax(αx) = α2Vmax(x)), and its unit level set is given by the
intersections of the ellipsoids defined by the matrices Qj , i.e LV max(1) = ∩E(Qj). While,
as far as the convex hull function is concerned the following properties holds: Vc is convex
positive homogeneous of degree 2, continuously differentiable and its unit level set is given
by the convex hull of the ellipsoids defined by the matrices Q−1j , i.e LV c(1) = co
E(Q−1j )
.
For this reason Vc is called convex hull function, and it can be alternatively described as
the convex hull of quadratics 12x
TQ−1j x.
Since for LDI asymptotic and exponential stability are equivalent, the above result can be
exploited to obtain general stability results similar to what reported for the other families
of Lyapunov candidates. In this respect, exploiting Th. 4.4.2, and the conjugacy between
Vc and Vmax = V ∗c , the main result concerning stability of (4.46) is here recalled
Theorem 4.4.3 Let positive definite matrices Qk ∈ Rn×n, k ∈ [1, l] be given to construct
Vmax and Vc defined in (4.63), (4.64) respectively.
• For γ ∈ R, if there exists δijk ≥ 0, j, k ∈ [1, l] such that
ATi Qk +QkAi ≤
l∑
j=1
δijk(Qj −Qk)− γQk ∀k ∈ [1, l] (4.69)
then ∀x ∈ Rn, A ∈ co Ai it holds: ∂V T
maxAx ≤ −γVmax.
• For γ ∈ R, if there exists δijk ≥ 0, j, k ∈ [1, l] such that
QkATi +AiQk ≤
l∑
j=1
δijk(Qj −Qk)− γQk ∀k ∈ [1, l] (4.70)
then ∀x ∈ Rn, A ∈ co Ai it holds: ∂V T
c Ax ≤ −γVmax.
It’s further to remark that inequalities (4.69), (4.70) are not equivalent since they provide
only sufficient conditions for Vmax, Vc to be Lyapunov functions. In other words, if one
of the two condition is not satisfied, the other can be checked, this doubles the number of
tools that can be exploited to analyze and estimate the stability region of LDIs. When
i = 2, by using S-procedure, it’s possible to show that condition (4.69) becomes also
necessary (see [75] pag. 73).
In order to express the set inclusions for the local validity of the PLDI (4.10) associated
to saturate systems, the following properties are introduced ([92])
95
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
Lemma 4.4.4 Let Hu ∈ Rm×n and L(Hu) := x| Hu,wx ≤ 1, w ∈ [1,m], then
• LV c(1) ⊂ L(H) ⇔ 2HTq,w ∈ LV c∗ ∀ w ∈ [1,m]
• LV max(1) = LV c∗(1) ⊂ L(Hu) ⇔ 2HTu,w ∈ LV c(1) ∀w ∈ [1,m]
Proof Vc and V ∗c = Vmax are positive definite and positive homogeneous of degree two,
hence they induce a pair of polar norms: ‖x‖c = (2Vc(x))1/2, ‖x‖∗c = (2V ∗
c (x))1/2. It
can be verified that for any z, and any δ > 0, |zTx| ≤ 1 for all ‖x‖c ≤ δ if and only if
‖z‖∗c ≤ 1/δ. Applying this fact with δ =√2 and z = HT
u,w yields |Hq,wx| ≤ 1 for all
Vc(x) ≤ 1, if and only if V ∗c (Hu,w) = 4V ∗
c (HTu,w) ≤ 1. This prove item one of the lemma,
the second item can be showed similarly.
Based on the above lemma and Th. 4.4.3 the following invariance conditions for the level
sets LV c(1), LV max(1) can be given
Theorem 4.4.5 Let Vc be the convex hull function formed by matrices Qk as defined in
(4.64), and take γ > 0, if there exist matrices Cu, Hu and numbers δijk ≥ 0 such that
Qk(A+B(DiCu +D−i Hu)
T + (A+B(DiCu +D−i Hu)Qk ≤ δijk
l∑
k=1
δijk(Qj −Qk)− γQk
2HTu,wQjHu,w ≤ 1
∀i ∈ [1, 2m], j, k = 1, . . . , l, w = 1, . . . ,m
(4.71)
then for the saturated closed-loop system (4.1) it holds
∂V Tc (Ax+Bsat(Cux)) ≤ −γVc ∀x ∈ LV c(1) (4.72)
Theorem 4.4.6 Let V ∗c = Vmax be the max quadratic function formed by matrices Qk as
defined in (4.63), and take γ > 0, if there exist matrices Cu, Hu and numbers δijk ≥ 0
such that
(A+B(DiCu +D−i Hu)
TQk +Qk(A+Bp(DiCu +D−i Hu) ≤ δijk
l∑
k=1
δijk(Qj −Qk)− γQk
2Hu,w ∈ LV c
∀i ∈ [1, 2m], j, k = 1, . . . , l, w = 1, . . . ,m
(4.73)
then for the saturated closed-loop system (4.1) it holds
∂V Tmax(Ax+Bsat(Cux)) ≤ −γVmax, ∀x ∈ LV c∗(1) = LV max(1) (4.74)
the above theorems can be proved by combining Th. 4.4.3 lemma 4.4.4 and the results
of Th. 4.2.4. Following the same reasoning made in 4.3 the invariant sets LV c(1) and
96
4.4. Improvements via non-quadratic Lyapunov functions
LV max(1) can be used to obtain an estimation of the system domain of attraction, maxi-
mizing their size by formulating the following two problems
supQk,γ,δijk,Cu,Hu
α
s.t. 4.71
αXR ⊂ LV c
γ > 0, δijk ≥ 0, Qj > 0 ∀ i, j, k
(4.75)
supQk,γ,δijk,Cu,Hu
α
s.t. 4.73
αXR ⊂ LV max
γ > 0, δijk ≥ 0, Qj > 0 ∀ i, j, k.
(4.76)
Here the stabilizability and feedback synthesis problem has been concerned, stability anal-
ysis trivially follows if matrix Cu is given. Noting that conditions in Lemma 4.4.4 can be
respectively expressed as the inequalities[
1 Hq,w
HTq,w
∑lk=1 λkQk
]
≥ 0
[12 Hu,wQk
QkHTu,w Qk
]
≥ 0
(4.77)
similarly to what in (4.3), the above problems can be cast into BLMIs as long as the shape
reference set XR is convex. If a single element Q is considered, it is straightforward to
verify that the quadratic stability LMI problem (4.18) is recovered.
Recently it has been proved in ([26]) that composite quadratic functions are universal for
polytopic LDIs, hence the stability and stabilizability conditions obtained before can be
showed to be also necessary if the numberl of considered quadratics is let to be any integer,
i.e l > n, where n is the order of systems defining the LDI. Although conservatism in the
Lyapunov analysis would be completely eliminated, the computational burden increases
if a larger set of quadratics is used to construct the composite funcitions, even it is in
general less demanding with respect to a polyhedral function based approach.
As mentioned, all the above results can still be enhanced letting the feedback law to be
nonlinear; in [24] a nonlinear synthesis approach has been proposed for the stabilization
problem of LDIs in the form (4.47). Since the convex hull quadratic function is endowed
with continuous differentiability, it is preferred for synthesis purposes, as it allows to ensure
continuity of the resulting control law. The main result discussed in [24] is here reported
Theorem 4.4.7 Let Vc be the convex hull function defined by matrices Qk = QTk , and
λ > 0. Suppose there exist matrices Yk, and numbers δijk ≤ 0, j, k ∈ [1, l], such that
QkATi +AiQk +BiYk + Y T
k Bi ≤l∑
j=1
δijk(Qj −Qk)− γQk (4.78)
97
Chapter 4. CONTROL OF LINEAR SATURATED SYSTEMS
then a stabilizing nonlinear feedback law can be constructed as follows. For each x let
λ∗(x) = argminλ∈ΓlxT(∑l
k=1 λkQk
)−1x, and denote
Y (λ∗(x)) =l∑
k=1
λ∗kYk, Q(λ∗) =l∑
j=k
λ∗kQk, K(λ∗) = Y (λ∗)Q(λ∗)−1 (4.79)
defining k(x) = K(λ∗(x))x, it follows
max∇Vc(x)T (Aix+Bik(x)) ≤ −βVc(x) ∀x (4.80)
for some β > 0.
The above result can be easily specified for saturated linear systems represented by the
local LDI (4.10), extending the claim of Th. 4.4.5;
Theorem 4.4.8 Let Vc be the convex hull function formed by matrices Qk as defined in
(4.64), and take γ > 0, if there exist matrices Yk, Zk and numbers δijk ≥ 0 such that
AQk +QkAT +BDiYk + (BD−
i Yk)T +BD−
i Zk + (BD−i Zk)
T ≤l∑
j=1
δijk(Qj −Qk)− γQk
[12 Zk,w
ZTk,w Qk
]
≥ 0
∀i ∈ [1, 2m], j, k = 1, . . . , l
(4.81)
then
∂V Tc (Ax+Bsat(Cux)) ≤ −γVc ∀x ∈ LV c(1). (4.82)
According to Th. 4.4.7, the nonlinear feedback law can then be recovered as Cu(λ∗(x)) =
Y (λ∗(x))Q(λ∗(x))−1, Hu(λ∗(x)) = Z(λ∗(x))Q(λ∗(x))−1. The method seems promising
to cope with practical problems and effectively extend the results obtained by simple
linear controllers, however the computational burden needed to compute the nonlinear
gain matrix is significantly increased. Indeed, beside a non convex BMI problem, a convex
minimization problem providing λ∗(x) has to be solved on-line since it depends on the
current state x.
98
Chapter 5
Control Design for Power
Converters fed by Hybrid Energy
Sources
This chapter addresses saturated control of a class of power systems driven by
battery/supercapacitor hybrid energy storage devices. The power flow from the
battery and the supercapacitor to the electrical load is actively controlled by two
bidirectional buck-boost converters. The LDI description presented in chapter 4
is exploited to deal with the resulting multiple saturated inputs and bilinear state-
space model. Stability and performance are optimized, casting the control design
problem into a numerically efficient problem with linear matrix inequalities.
5.1 Introduction and motivation
The battery/supercapacitor hybrid energy storage systems are widely used in electric,
hybrid and plug-in hybrid electric vehicles, and have received a considerable interest in
the specific literature (see [93],[94], [95],[96] for a comprehensive overview). Under the
fast growth of renewables, they have also found applications in wind systems [97], [98],
photovoltaic systems [99], and microgrids [100].
It is generally accepted that combining different types of energy storage devices can pro-
vide several advantages over using only one type of such devices alone [101]. In this respect
batteries and supercapacitors are commonly combined to obtain a system having both the
high energy density of the batteries and the high power density of the supercapacitors.
Such a combination is able to provide very high current to the load in a short period of
time, while maintaining a safe discharging current from the batteries. This strategy would
extend the life time of the batteries without sacrificing the performance of the whole sys-
tem.
The main functions of supercapacitors in such systems are to provide high currents during
hard transients (such as motor start), absorbing possible wind/solar power excess, and
99
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
storing energy from regenerative braking. To realize these functions, effective strategies
are required to allocate current flow among the different energy storage devices. The sim-
plest way is to connect the battery and the supercapacitor in parallel. This earlier method
has an obvious limitation since the currents from the power sources can not be controlled.
To actively steer system currents, especially the current from the battery, dc-dc convert-
ers are needed to connect the power sources and the load or DC bus. There are various
configurations and control strategies to implement active current control, as summarized
in [95],[102],[96]. In some configurations [103], [104], [101], a single dc-dc converter is used
to connect the battery and the supercapacitor, however the most commonly used config-
uration consists of two bidirectional buck-boost converters, each driven by the battery or
supercapacitor. The outputs of the two dc-dc converters are connected in parallel to the
load or DC bus, see Fig. 5.1. Such a configuration has been considered, for example, in
[105], [102], [106], [100], where different control strategies have been proposed to actively
steer the current flow from the battery and the supercapacitor. A common strategy is to
use a certain energy management algorithm to determine a reference current that is needed
from the battery or the supercapacitor, then use a simple decentralized PI control on each
dc-dc converter to track the respective reference current. Such a strategy assumes ideal
converters, disregarding the power loss in the circuit elements, furthermore the coupling
between the power converters is not considered. Finally saturation of the converter control
inputs and system bilinear terms are usually discarded in the regulators design procedure
which considered a linearized system around the predefined working point. As a result a
very small stability region of the actual nonlinear system can be ensured by means of this
techniques. As mentioned many times, power electronic systems are expected to robustly
work in a considerable wide range of situations which can bring them to drift from the
predefined equilibrium point, hence it would be profitable to guarantee a wide stability
region and a certain degree of performance over multiple conditions. In addition, on a
practical viewpoint, even if stability is preserved, the output power in the load may not
be the desired value and the tracking performance may not be satisfactory.
Here the typical configuration with two bidirectional dc-dc converters driven by corre-
sponding battery and supercapacitor as in Fig. 5.1 is considered; following the approach
proposed in [37] first the state-space description for the whole system is formally derived
in order to enlighten the couplings among circuit variables dynamics in the two converters
connected to the common load. Then the resulting bilinear saturated model is then de-
scribed with linear differential inclusions (LDIs) with four vertex. Based on this polytopic
description, the method discussed in ch. 4 and already applied to a single input power
converter in [107], is extended to design a feedback control law for the considered MIMO
system via optimization algorithms.
The main control objective can be regarded as a reference tracking for some variables
such as battery current, supercapacitor current, load voltage and load current. Since the
hybrid energy storage system has two control inputs (the duty cycles) of the two dc-dc
converters, it can track references for two circuit variables. Instead of choosing the battery
100
5.2. State-space averaged model
Re RL1 L1
S1
C2
E D1
Load
S2
S3
S4D2
L2RL2Ru
C1
Cu
CoR
Figure 5.1: Parallel topology of the buck-boost converters.
current and the supercapacitor current, commonly adopted in the specialized literature,
here the battery current and the voltage of the load are considered as controlled outputs.
This choice is motivated by the fact that the supercapacitor is playing a supporting role
and it can supply or absorb almost any current as needed, while the state-of-health of the
battery (which depends on its charging/discharging profiles) and the performance at the
load side are of high priority for the above mentioned typical applications.
The chapter is organized as follows. In Section 5.2 the averaged system dynamics are
formally derived, in Section 5.3 the system bilinear terms and saturation nonlinearity are
described by mean of a PLDI, based on this characterization a robust control solution,
maximizing the system tracking domain with ensured convergence rate is designed in a
similar fashion of what reported in 4.3.3. In 5.4 formal stability results are improved
by refining the system bilinearity description via a piecewise linear differential inclusions,
and associating the resulting representation with the class of piecewise quadratic Lya-
punov functions discussed in 4.4.1. Section 5.5 ends the chapter with simulation and
experimental results obtained on a reduced scale hybrid system.
5.2 State-space averaged model
In this section, the state-space averaged model for the hybrid energy storage system is
derived, following the well-known averaging method initiated by Middlebrook in [108].
As mentioned the the circuit topology of the hybrid energy storage system consists of
two standard bi-directional buck-boost converter connected in parallel at the load side,
and fed by a battery and a supercapacitor respectively. For semplicity of presentation,
the battery and the supercapacitor are described with very simple models which only in-
clude parasitic series resistances Re and Ru, respectively. More comprehensive models of
these devices ([109]) can be considered. However, as will be explained later, this would
will only increase the considered system order, without affecting the crucial feature of
system dynamics. Hence higher order dynamics related to battery and supercapacitors
modeling can be easily added “plugged-in” the proposed framework to improve enhance
real systems performances. Here a simple resistive load is considered, anyway, with small
variations, it can be replaced by more realistic load topologies, e.g as a an inverter DC-bus.
101
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
Circuit operating modes
Duty cycle S1 S2 S3 S4
P1 On Off On Off
P2 On Off Off On
P3 Off On Off On
P4 Off On Off On
Table 5.1
i1
vuCu
E C1
C2
Co
Re
Ru
Ron3
Ron1 R
RL1
RL2
L1
L2
v1
v2
vo
i2
Figure 5.2: Equivalent circuit for mode P1.
5.2.1 State space description for 4 operational modes
The two MOSFETs S1 and S2 operate synchronously, when one is off, the other is on, the
same applies for S3 and S4, thus four operating modes for the switching circuit can be
considered, as shown in Table 5.1. For the sake of simplicity the two buck-boost converters
are assumed to be operated at the same switching frequency. To obtain an averaged
model, each mode is considered separately, then a weighted average of the obtained the
state-space descriptions is performed with the durations of each possible condition, as
weighting coefficients. As regards the operating mode P1, S1 and S3 are on, while S2
and S4 are off. In this mode, S1 and S3 can be simply modeled as resistors Ron1 and
Ron3, respectively. The on resistance for S2, S4 are denoted Ron2 and Ron4, respectively,
as reported in equivalent circuit for this mode drawn in Fig. 5.2. Let v1, v2, vu, vo be
the capacitor voltages and i1, i2 be the inductor currents (see the assignment in Fig. 5.2).
Denote the state vector as ζ = [v1 v2 vu i1 i2 vo]T , then the state-space description can
be obtained by applying Kirchoff’s voltage law and Kirchoff’s current law. The following
102
5.2. State-space averaged model
matrices are defined to compact the notation for the 4 operating modes:
A11 =
− 1ReC1
0 0
0 − 1RuC2
1RuC2
0 1RuCu
− 1RuC2
, A12 =
− 1C1
0 0
0 − 1C2
0
0 0 0
, A21 =
1L1
0 0
0 1L2
0
0 0 0
A22,1 =
−RL1+Ron1L1
0 0
0 −RL2+Ron3L2
0
0 0 − 1RC0
, Ap1 =
[
A11 A12
A21 A22,1
]
(5.1)
and Be = [ 1ReC1
0 0 0 0 0]T . By these definitions, the state-space description for
operating mode P1 results:
ζ = Ap1ζ +BeE (5.2)
where E is the battery voltage. Similar considerations apply for the other modes, whose
circuit can be drawn similarly to what in 5.2; as regards P2 (S1, S4 on, S2, S3 off), defining
A22,2 =
−RL1+Ron1L1
0 0
0 −RL2+Ron4L2
− 1L2
0 1Co
− 1RCo
, Ap2 =
[
A11 A12
A21 A22,2
]
(5.3)
yields the state space description:
ζ = Ap2ζ +BeE. (5.4)
While as concern modes P3, P4, denoting
A22,3 =
−RL1+Ron2L1
0 − 1L1
0 −RL2+Ron3L2
01Co
0 − 1RCo
, A22,4 =
−RL1+Ron2L1
0 − 1L1
0 −RL2+Ron4L2
− 1L2
1Co
1Co
− 1RCo
(5.5)
and
Ap3 =
[
A11 A12
A21 A22,3
]
, Ap4 =
[
A11 A12
A21 A22,4
]
(5.6)
the following state-space descriptions are respectively obtained for operation mode 3,4
ζ = Ap3ζ +BeE (5.7)
ζ = Ap4ζ +BeE (5.8)
5.2.2 Averaged model for the open loop system
On the basis of the previously obtained state-space representation the system averaged
dynamics can be expressed as
˙ζ =
(4∑
i=1
diAPi
)
ζ +BeE, (5.9)
103
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
where ζ denotes the state averaged vector over one switching period T , while di represents
the duration for each operating mode normalized over T . It is clear that the average
of the matrices can be carried out by each element, such as APi(j, k), i = 1, 2, 3, 4. This
element-wise averaging may be convenient when APi’s have good structures. However, the
determination of di’s is a tedious procedure, since they depend not only on the converters
duty cycles D1 and D2, associated with the on state of S1, S3 respectively, but also
on the time difference between the turning-on of instants of the two switches. Denote
this time difference as δT (δ ∈ [0, 1)), i.e.., S3 is turned on after S1 is turned on by
δT . Now assume that δ = 0, i.e.., S1 and S3 turning on instants are synchronous. If
D1 > D2, then d1 = D2, d2 = D1 − D2, d3 = 0, d4 = 1 − D1, while if D1 < D2, then
d1 = D1, d2 = 0, d3 = D2 − D1, d4 = 1 − D2. The situation would be more involved
if δ 6= 0, since the expressions for d′is would depend on the relative size of δ,D1 and
D2, as a result the analysis should be broken down into several cases. Fortunately, the
expressions for di’s it’s not needed in this case, thanks to the properties of the matrices
Api, i = 1, 2, 3, 4. In fact, it turns out that the resulting averaged model does not rely
on δ and has only one expression for all possible cases. First note that all the 4 APi’s
have the same blocks A11, A12 and A21 (see (5.1), (5.3), (5.6)), thus they are these blocks
are not affected by the averaging procedure, and only the nonzero elements of the block
A22,i, i = 1, 2, 3, 4 need to be considered. Let begin with A22,i(1, 1) first; by (5.1), (5.3),
(5.5) it can be verified that the average of A22,i(1, 1), i = 1, 2, 3, 4 over one switching
period is A22,avg(1, 1) = −RL1+D1Ron1+(1−D1)Ron2
L1. Similarly, the average of A22,i(2, 2) is
A22,avg(2, 2) = −RL2+D2Ron3+(1−D2)Ron4
L2. As concerns A22,i(1, 3), it is 0 for modes P1, P2
(S1 on) and equals − 1L1
for modes P3, P4 (S1 off). Hence A22,avg(1, 3) = − 1L1
(1 − D1).
Similar arguments can be used to average the other elements;
A22,avg(2, 3) = − 1
L2(1−D2), A22,avg(3, 1) =
1
Co(1−D1)
A22,avg(3, 2) =1
Co(1−D2), A22,avg(3, 3) = − 1
RCo.
(5.10)
Combining the above results and defining
W0 =
−RL1+Ron2L1
0 − 1L1
0 −RL2+Ron4L2
− 1L2
1Co
1Co
− 1RCo
, W1 =
Ron2−Ron1L1
0 1L1
0 0 0
− 1Co
0 0
W2 =
0 0 0
0 Ron4−Ron3L2
1L2
0 − 1Co
0
(5.11)
the average of A22,i’ can be expressed in the compact form
A22,avg =W0 +W1D1 +W2D2. (5.12)
Finally, to describe the averaged model, denote
A0 =
[
A11 A12
A21 W0
]
, A1 =
[
0 0
0 W1
]
, A2 =
[
0 0
0 W2
]
(5.13)
104
5.2. State-space averaged model
then the averaged model is
˙ζ = (A0 + A1D1 + A2D2)ζ +BeE (5.14)
the same method can be applied if more comprehensive models for the battery and the
supercapacitor are considered. The only difference will be a higher dimension of ζ due
to the additional variables associated to the battery and the supercapcitor parasitic ca-
pacitor voltage dynamics (see [109]). Assume that the new voltage variables are stacked
on top of the original state ζ. Since the capacitors in the battery/supercapacitor models
are not directly connected to the MOSFETs, like Cu, C1, C2, they will not affect A22,i’s.
Accordingly, the corresponding A11, A12 and A21 will be fixed but with higher dimensions.
Thus the structure of the averaged model is the same. In general system (5.14), can be
completed with a controlled output equation y = Cζ +DE, where y is any combination
of two averaged state variables. As mentioned, here the tracking of desired references for
the battery current ib and the load voltage vo is considered; hence the output equations
specialize to
ib =
[
− 1
Re0 0 0 0 0
]
ζ +1
ReE
vo = [0 0 0 0 0 1]ζ .
(5.15)
Since C1 and C2 are small filter capacitors, in the following i1 will be considered in place
of ib for convenience.
Model (5.14) can be used for simulation of the open-loop system under constant duty
cycles D1 and D2, as well as for the closed-loop system under particular control strategies.
However, it is not suitable for steady-state analysis or control design. For a power converter
driven by ideal voltage sources, a steady state will be reached (usually very quickly, e.g.,
within a few milliseconds) when a constant duty cycle is applied and the steady state can
be used to determine the corresponding equilibrium point for the system variables. The
steady state can be easily computed by setting ˙ζ = 0 in (5.14). When a supercapacitor is
used as a power source/sink, it would take much longer (e.g., from many seconds to a few
minutes) to reach a steady state. Furthermore, this steady state is generally not a useful
or a desired operating condition for the considered applications. The reason is that, at a
steady state, dvu/dt = 0, this implies that the supercapacitor is not supplying/absorbing
current and thus not assisting the power system. However, a nominal working condition,
which has to be a steady state (or an equilibrium point), needs to be considered for control
design (stabilization or tracking) purposes, in order to derive a perturbation model. To
handle this situation, a similar approach to what presented in 3.2 for Shunt Active Filters
can be applied; relying on a suitable supercapacitor sizing, capable of providing a time-
scale separation between the supercapacitor dynamics and the remaining state variables,
allows to exploit singular perturbation theory arguments and replace Cu with an ideal
voltage source whose value is varying “slowly” within a certain range. In this respect,
a robust feedback controller will be derived for a lower order “fast” subsystem, obtained
disregarding the supercapacitor voltage dynamics, but able to handle the the uncertain
and varying ideal voltage source value.
105
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
When the supercapacitor Cu is replaced with an ideal voltage source Eu, a 5th order
averaged model can be derived with similar approach as in last section. Denote the state
variable and its average as ξ = [v1 v2 i1 i2 vo]T , ξ, and Ve = [E Eu]
T , then the 5th
order averaged model can be written as
˙ξ = (A0 + A1D1 + A2D2)ξ + BeVe (5.16)
where matrices A0, A1, A2 can be easily derived by those reported in (5.13) neglecting vu.
5.3 Saturated controller design for robust output tracking
As mentioned in the previous section, stabilization for the original hybrid energy storage
system as described in Fig. 5.1 and the 6th-order averaged model (5.14) is not a meaningful
problem since no steady state (or equilibrium point) is a desired operating condition as
it would imply null current provided/drained by the supercapacitor. The system has two
control inputs D1 and D2 then, in principle, the reference for an arbitrary two-dimensional
output, y = Cξ where C is a matrix of two rows, can be tracked. However, due to the
control input hard constraints D1, D2 ∈ [0, 1], the following facts should be realized:
• For the 5th order model where the supercapacitor is replaced with an ideal voltage
source Eu, for each output reference yref , there is a certain range for the voltages
pair (E,Eu) where tracking is possible. On the other hand, for a given range of
(E,Eu), there is a certain set of yref which can be feasibily tracked.
• For the original 6th order system with supercapacitor, any tracking can only last
for a finite time period, beyond which the supercapacitor voltage will drop (or rise)
out of the range where yref can be tracked. If some a priori information about
the power required by the load is available, e.g a benchmark periodic load profile
is known, then a suitable sizing of the supercapacitor, such that its voltage never
drifts from a predefined range during the load switching cycle, can be carried out, in
a similar fashion to what discussed in 3.2 as regards the DC-bus capacitor of Shunt
Active Filters. In addition, the battery reference current can be augmented with a
term, given by a slow control loop, devoted to keep the supercapacitor averaged volt-
age value unchanged over a load switching cycle. This can be achieved by exploiting
the averaging theory framework as in 3.4, or by constrained convex optimization
arguments ([102], [96]).
Here the focus is put on the fast varying dynamics control, neglecting the supercapac-
itor averaged voltage regulation, and regarding its slow variations as an uncertainty
to be managed by the feedback controller.
106
5.3. Saturated controller design for robust output tracking
5.3.1 Converting the tracking problem to a stabilization problem
Based on the previous considerations, the 5th order model (5.16) is considered for control
design, selecting y = [i1 vo]T as the controlled output, it can be completed as follows
˙ξ = (A0 + A1D1 + A2D2)ξ + BeVe
y = Cξ(5.17)
where Ve = [E Eu]T and
C =
[
0 0 1 0 0
0 0 0 0 1
]
. (5.18)
Before designing a control law, a suitable nominal operating condition has to be chosen
for the variables: R0, Ve0 = [E0 Eu0]T D10, D20, and ξss,0 satisfying
ξss,0 = −(A0 + A1D10 + A2D20)−1BeVe0. (5.19)
Given a pair (E,Eu), a reference yref is feasible if there exist D1, D2 ∈ [0, 1] such that
Cξss,0 = yref where ξss,0 is the solution of the above equilibrium equation. Actually, it’s
worth to remark that due to the nonlinear nature of (5.19) if ibref , vo,ref are plugged
into ξss,0 in (5.19), and the remaining states and the two duty cycles are considered
as variables, some bifurcation-like behaviors, not unusual for power electronic systems
([110]), can occur. For the considered application, if the parasitic resistance RL1, RL2,
Re,Ru are accounted in the duty cycle calculation, for possibly feasible references cannot
be associated with D10, D20 belonging to the real field (see [37]).
The next natural step is to derive a perturbation model around a given nominal condition.
To this aim define x = ξ − ξss,0, y = y − Cξss,0, u1 = D1 − D10, u2 = D2 − D20 and
u = [u1 u2]T . Denote also
A0 = A0 + A1D10 + A2D20, B = [A1ξss,0 A2ξss,0] (5.20)
then, by plugging ξ = x+ ξss,0, D1 = u1+D10 and D2 = u2+D20 into (5.17) and applying
(5.19), yields the following perturbation model:
x = A0x+ A1xu1 + A2xu2 +Bu+ Be(Ve − Ve0) + A0ξss,0
y = Cx.(5.21)
where A0 accounts for a different load resistor from the nominal value. A feedback law can
be designed to stabilize the origin of the system (5.21) under the nominal condition where
Ve = Ve0 and for nominal load resistance value R0, however, it should be expected that,
on a real circuit, Ve 6= Ve0 as the battery and the supercapacitor voltage values are always
changing. Furthermore, the desired value for the output is also changed frequently and
the load resistance is not a constant in the most of the applications. To achieve robust
reference tracking in the presence of uncertainties a standard integral augmentation is
performed, defining
xa =
∫
(y − r)dt =
∫
(Cx− yref )dt (5.22)
107
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
where r(t) is a generic desired reference profile superposed to the nominal condition yref .
Then, defining the corresponding augmented state xw :=
[
x
xa
]
, that includes the pertur-
bations of the voltages of three capacitors C1, C2, Co, the two inductor currents, and two
integrator outputs, along with matrices
A =
[
A0 0
C 0
]
, Ab1 =
[
A1 0
0 0
]
, Ab2 =
[
A2 0
0 0
]
B =
[
B
0
]
, g =
[
Be(Ve − Ve0) + A0ξss,0
−yref
]
where the zero blocks have compatible dimension, the following augmented error dynamics
are obtained
xw = Axw + Ab1xwu1 + Ab2xwu2 + Bu+ g. (5.23)
Therefore the control design objective can be stated as
• Design a feedback control law u = f(xw) which stabilizes (5.23) at the origin with
a large stability region, under the nominal condition g = 0, and under the control
inputs constraint D1, D2 ∈ [0, 1].
It’s further to notice that, if the system drifts away from the nominal working con-
dition and goes to another equilibrium point, each state variable, in particular, the
integral xa, will still reach a steady state. This means that y− yref must go to zero
and the output y is regulated to the desired value yref .
5.3.2 State feedback law design via LMI optimization
In what follows, a stabilizing feedback for (5.23) under input constraint is presented, by
adopting and extending the techniques presented in 4.3. Beside input constraints, the
system is bilinear, here also bilinearity is handled by means of a polytopic inclusion,
following the philosophy already proposed in [109], [107], for single input converters.
Constraints Dj ∈ [Djmin, Djmax] ⊂ [0, 1], j = 1, 2 can be trivially mapped into the
variables uj , j = 1, 2, that is Djmin − Dj0 ≤ uj ≤ Djmax − Dj0. Denoting umj =
Dj0 −Djmin and upj = Djmax −Dj0, the constraints can be expressed as
− umj ≤ uj ≤ upj , j = 1, 2. (5.24)
These input limitations can be clearly enforced via a decentralized saturation function
sat(u) = [sat(u1) sat(u2)]T similar to what in (1.1), i.e.
sat(uj) =
upj if uj > upj
uj ifuj ∈ [−umj , upj ]
−umj if uj < −umj
(5.25)
108
5.3. Saturated controller design for robust output tracking
Considering a simple saturated state feedback law u = sat(Kx), where K =
[
K1
K2
]
∈ R2×7,
the following closed loop system is derived
xw = (A+ Ab1sat(K1xw) + Ab2sat(K2xw))xw + Bsat(Kxw). (5.26)
The nonlinear terms Ab1sat(K1xw), Ab2sat(K2xw) can be described (with some conser-
vatism) with a polytopic inclusion, according to the “global linearization” principle (see
[75] ch.4). In particular, since umj ≤ Kj ≤ upj (where Kj denotes the jth raw of matrix
K), defining
A1 = A− um1Ab1 − um2Ab2
A2 = A− um1Ab1 + up2Ab2
A3 = A+ up1Ab1 − um2Ab2
A4 = A+ up2Ab1 + up2Ab2
the following four vertices inclusion characterizes system (5.26)
xw ∈ co
Ai + Bsat(Kxw)
xw, i = 1, . . . , 4. (5.27)
Thus the same approach applied to describe saturated linear system in (4.2) can be applied
to the saturated inclusion (5.27), obtaining the PLDI
xw ∈ co
(Ai + BDjK ++BD−j Hu)
xw, i = 1, . . . , 4, j = 1, . . . , 4 (5.28)
which is similar to (4.10). Therefore all the LMI-based optimization methods presented
in 4.3 can be exploited to design a feedback control law meeting the system specifications.
As mentioned, for this application, the main objective is to ensure a wide stability region
of the desired working point, however a certain degree of system responsiveness is usually
required. Recalling the considerations reported in 4.3.3 this request can be mapped into
a convergence rate request for the closed-loop system, which, due to the limited control
authority, has to be suitably balanced with the need of a large basin of attraction. As a
consequence, taking a classic quadratic control Lyapunov candidate V (xw) = xTwPxw, the
following problem, similar to what in 4.45 is formulated
infQ>0,Y,γ
γ
s.t. AiQ+QATi + BY + Y T BT < −2ηQ, i = 1, . . . , 4
[
min(u2mj , u2pj) Yj
Y Tj Q
]
≥ 0, j = 1, 2
[
Q I
I γ
]
≥ 0
for a given convergence rate η the problem is an EVP.
By result of Th. 4.2.4 and convexity of the matrices set
Ai
4
i=1, it is straightforward to
prove that the inequality constraints in (5.29) ensures quadratic stability of the inclusion
109
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
Circuit parameters
Inductor L1 [µH] 680
Battery model resistor Re [Ω] 0.04
Inductor parasitic resistor RL1 [Ω] 0.25
Battery side filter capacitor C1 [mF] 1
Inductor L2 [µH] 39
Supercapacitor model resistor Ru [Ω] 0.011
Inductor parasitic resistor RL2 [Ω] 0.114
Capacitor C2 [m F] 0.22
Supercapacitor Cu [F] 116
Mosfet on-resistance Ron [Ω] 0.021
Load side capacitor C [mF ] 1
Table 5.2
(5.28) inside the ellipsoid E(Q−1), which is in turn contractive invariant. The scaled unit
ball δI, with δ = 1√γ has been selected as shape reference set with respect to compare
the ellipsoid size, providing an estimation of the basin of attraction. Finally, the optimal
state feedback gain matrix K can be recovered as K = Y Q−1.
5.3.3 Numerical result for an experimental setup
In order to motivate the extended stability analysis provided in 5.4, the results of the
control design method reported in the previous section are presented for an experimental
system constructed according to the topology reported in Fig.5.1. The circuit fixed pa-
rameters are provided in Tab. 5.2, while the load resistance R is variable. In the tests
reported in 5.5, it is switched between 2Ω (heavy load) and 200Ω (light load). The control
design has been carried out considering R0 = 2Ω as the nominal condition, while the
nominal battery voltage has been chosen as E0 = 6V and the nominal voltage for the
ideal voltage source in place of the supercapacitor is selected as Eu0 = 7V . The nominal
reference output vector is defined as y0 = [3 10]T . By solving (5.19) for these numerical
data, it turns out that the unique duty cycles paie (D1, D2) that produces this nominal
output is (D10, D20) = (0.4933, 0.3822), and the corresponding steady state for the 5th-
order averaged model is ξss,0 = [5.88 6.938 3 5.632 10]T . For the duty cycle, the
restriction D1, D2 ∈ [0.2, 0.8] is imposed to take into account realistic converters, which
cannot operate over the full range of the duty cycle values, thus the corresponding bounds
on u1, u2 are: um1 = 0.2933, up1 = 0.3067, um2 = 0.1822, up2 = 0.4127. By solving
(5.29) for these parameters and with η = 5, 25, by means of the standard MATLABTM
solver “mincx” (which minimizes linear objective under LMI constraints), the following
110
5.4. Stability and tracking domain analysis via piecewise quadratic Lyapunov functions
two matrices are obtained
K5 =
[
−0.002 0 −0.037 0 0.02 −1.16 −0.57
0 0 −0.002 0 −0.006 1.8 −5.08
]
K25 =
[
−0.004 0 −0.08 −0.001 0.04 −6.6 −4.11
0 0 −0.008 −0.002 −0.03 10.11 −23.9
] (5.29)
and the problem optimal values are respectively γ = 12.21, 321.1, corresponding to α =1√γ = 0.288, 0.056. Computing the maximal scaled unit ball, contained in the invariant
ellipsoid, as α×xw ∈ R
T : xTwxw ≤ 1, it turns out that very small domains of attraction
are ensured. Such estimates are not very useful considering the possible initial conditions
of the circuit variables (which can be evaluated by sweeping D1, D2 over the admissible
range and founding the corresponding steady state by 5.19). The reason for such an high
conservative results is twofold; bilinear terms are pretty roughly approximated by the four
vertices inclusion (5.27), and, as remarked in 4.4, quadratic stability tools can be very
limiting when applied to nonlinear systems.
In the the next section two countermeasures will be taken in the same fashion of the
work [107] where a single boost converter was considered: first the system bilinearity is
described by means of a piecewice LDI model, in order to obtain a closer description of the
original system, then, in the light of the methods presented in 4.4.1, a piecewise Lyapunov
candidate will be considered in order to extend the domain of attraction estimation.
5.4 Stability and tracking domain analysis via piecewise
quadratic Lyapunov functions
Here the objective is to extend the stability results obtained in the previous section by
means of an LDI representation and quadratic Lyapunov candidates. Moreover the sta-
bility region of non nominal working condition, caused for example by battery and super-
capacitor voltage variations, a different load resistance, or a desired reference term r 6= 0,
will be discussed considering a non null g in (5.23). Assuming that a linear feedback law
has been designed relying on the techniques presented in the previous section, the resulting
closed loop
xw = Axw + Ab1sat(K1xw)xw + Ab2sat(K2xw)xw + Bsat(Kxw) + g (5.30)
where B1, B2 are the rows of matrix B, can be described by means of piecewise LDIs, as
previously mentioned. In general, the stability properties of a generic working point xe
satisfying
0 = Axe + Ab1(K1xe)xe + Ab2(K2xe)xe + B(Kxe) + g (5.31)
need to be considered. Note that the admissible references r(t) and/or working conditions,
are those producing a vector g compliant with the system constraints, i.e. the above
equation needs to be satisfied for |Kjxe| ≤ min(umj , upj), j = 1, 2. Obviously for g =
0 ⇒ xe = 0 and the problem reduces to evaluate the domain of attraction of the nominal
111
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
working point defined in the previous section. To examine stability around a generic xe,
the following change of coordinates is defined: z = xw − xe. Then, subtracting (5.31)
from (5.30), after some computations (adding and subtracting B(Kxe) + Ab1xe(K1xe) +
Ab2xe(K2xe) + Ab1zK1xe + Ab2zK2xe) the dynamics of z are obtained
z = Arz+[Br1+Ab1z](sat(K1(z+xe)−K1xe)+[Br2+Ab2z](sat(K2(z+xe)−K2xe) (5.32)
where Ar = A + Ab1K1xe + Ab2K2xe, Br1 = B1 + Ab1xe, Br2 = B2 + Ab2xe and B1, B2
are the columns of B.
Following the approach presented in ([107], [109]), the focus is restricted to a polytopic
state space region bounded by the four hyperplanes, K1z = ±c1, K2z = ±c2. For sake
of simplicity we assume c1 = c2 = c, where c > upj −Kjxe, ∀j = 1, 2 and −c < −umj −Kjxe, ∀j = 1, 2. Thus the objective can be stated to search for the largest controlled
invariant set ( under the saturated law Kxw) contained in the above defined state space
subset, which is compactly expressed as Ic = z : |Kjz| ≤ c j = 1, 2.As mentioned, the first step is to improve the approximation accuracy of the bilinear
terms. For this purpose each control input component is associated with two sets of
N − 1 positive and negative scalars (in principle the numbers of positive and negative
coefficients can differ, here they are assumed equal to simplify the notation), aj1, . . . , ajN ,
bj1, . . . , bjN , such that: 0 < aj1 < aj2 · · · < ajN−1 < upj − Kjxe, 0 > bj1 < bj2 · · · <bjN−1 > −umj −Kjxe, j = 1, 2 and ajN = upj −Kjxe, bjN = −umj −Kjxe, ajN+1 = c,
bjN+1 = −c. Hence Ic can be partitioned into (2N +1)2 polytopes Ωij , i = 1, . . . , 2N +1,
j = 1, . . . , 2N + 1 defined as
Ω00 := z| b1,1 ≤ K1z ≤ a1,1 ∩ z| b2,1 ≤ K2z ≤ a2,1Ωi0 := z| a1,i ≤ K1z ≤ a1,i+1 ∩ z| b2,1 ≤ K2z ≤ a2,1Ω0j := z| b1,1 ≤ K1z ≤ a1,1 ∩ z| a2,j ≤ K2z ≤ a2,j+1Ωij := z| a1,i ≤ K1z ≤ a1,i+1 ∩ z| a2,j ≤ K2z ≤ a2,j+1 i, j = 1, . . . , N
Ωi0 := z| b1,i+1 ≤ K1z ≤ b1,i ∩ z| b2,1 ≤ K2z ≤ a2,1Ω0j := z| b1,1 ≤ K1z ≤ a1,1 ∩ z| b2,j+1 ≤ K2z ≤ b2,jΩij :=
z| b1,(i+1) ≤ K1z ≤ b1,i
∩ z| b2,j+1 ≤ K2z ≤ b2,j i, j = N + 1, . . . , 2N
Ωij := z| b1,i+1 ≤ K1z ≤ b1,i ∩ z| a2,j ≤ K2z ≤ a2,j+1 i = N + 1, . . . , 2N, j = 1, . . . , N
Ωij := z| a1,i ≤ K1z ≤ a1,i+1 ∩ z| b2,j+1 ≤ K2z ≤ b2,j i = 1, . . . , N, j = N + 1, . . . , N
(5.33)
where only Ω0 contains the origin. The geometric interpretation of the above partition is
drawn in Fig. 5.3
5.4.1 Piecewice LDI description
In view of the partition defined in (5.33), system (5.32) can be represented by a four
vertices LDI for each of the polytopic cells Ωij . Inside Ω0, similarly to (5.27), it can be
verified that the following LDI description holds
z ∈ co A0pp, A0pm, A0mp, A0mm z (5.34)
112
5.4. Stability and tracking domain analysis via piecewise quadratic Lyapunov functions
a11 a12 (-c,c)b11b12(-c,-c)
a21
a22
b21
b22
(c,c)(-c,c)
Ω00 Ω10 Ω20Ω30Ω40
Ω01
Ω02
Ω03
Ω04
Ω11
Ω12
Ω13
Ω14
Ω21
Ω22
Ω23
Ω24
Ω31
Ω32
Ω33
Ω34
Ω41
Ω42
Ω43
Ω44
Figure 5.3: Partition example of Ic with N = 2.
with A0pp = (Ar + Ab1a1,1+ Ab2a2,1+Br1K1+Br2K2), A0mm = (Ar + Ab1b1,1+ Ab2b2,1+
Br1K1+Br2K2), A0mp = (Ar+Ab1b1,1+Ab2a2,1+Br1K1+Br2K2), A0pm = (Ar+Ab1a1,1+
Ab2b2,1 + Br1K1 + Br2K2). While when i, j = N , saturation occurs at both inputs and
the LDI representation “collapses” to a single affine system defined as
z = ANN
[
z
1
]
,
ANN =
[
Ar + Ab1a1,N + Ab2b2,N Br1a1,N +Br2a2,N
01×N 0
]
.
(5.35)
Similar considerations can be made to obtain the system description inside the other cells;for i, j = 1, . . . , N − 1 define
Ai0pp =
[
(Ar +Ab1a1,i+1 + Ab2a2,1 +Br1K1 +Br2K2) 0
01×7 0
]
, Ai0mp =
[
(Ar +Ab1a1,i + Ab2a2,1 +Br1K1 +Br2K2) 0
01×7 0
]
Ai0pm =
[
(Ar +Ab1a1,i+1 + Ab2b2,1 +Br1K1 +Br2K2) 0
01×7 0
]
, Ai0mm =
[
(Ar +Ab1a1,i + Ab2b2,1 +Br1K1 +Br2K2) 0
01×7 0
]
A0jpp =
[
(Ar + Ab1a1,1 + Ab2a2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
, A0jmp =
[
(Ar + Ab1b1,1 + Ab2a2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
A0jpm =
[
(Ar + Ab1a1,1 + Ab2a2,j +Br1K1 +Br2K2) 0
01×7 0
]
, Aijmm =
[
(Ar + Ab1b1,1 + Ab2a2,j +Br1K1 +Br2K2) 0
01×7 0
]
Aijpp =
[
(Ar + Ab1a1,i+1 + Ab2a2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
, Aijmp =
[
(Ar + Ab1a1,i + Ab2a2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
Aijpm =
[
(Ar + Ab1a1,i+1 + Ab2a2,j +Br1K1 +Br2K2) 0
01×7 0
]
, Aijmm =
[
(Ar + Ab1a1,i + Ab2a2,j +Br1K1 +Br2K2) 0
01×7 0
]
(5.36)
113
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
while if i, j = N + 1, 2N − 1
Ai0pp =
[
(Ar + Ab1b1,i + Ab2a2,1 +Br1K1 +Br2K2) 0
01×7 0
]
, Ai0mp =
[
(Ar + Ab1b1,i+1 + Ab2a2,1 +Br1K1 +Br2K2) 0
01×7 0
]
Ai0pm =
[
(Ar + Ab1b1,i + Ab2b2,1 +Br1K1 +Br2K2) 0
01×7 0
]
, Ai0mm =
[
(Ar + Ab1b1,i+1 + Ab2b2,1 +Br1K1 +Br2K2) 0
01×7 0
]
A0jpp =
[
(Ar + Ab1a1,1 + Ab2b2,j +Br1K1 +Br2K2) 0
01×7 0
]
, A0jmp =
[
(Ar + Ab1b1,1 + Ab2b2,j +Br1K1 +Br2K2) 0
01×7 0
]
A0jpm =
[
(Ar + Ab1a1,1 + Ab2b2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
, A0jmm =
[
(Ar + Ab1b1,1 + Ab2b2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
Aijpp =
[
(Ar + Ab1b1,i + Ab2b2,j +Br1K1 +Br2K2) 0
01×7 0
]
, Aijmp =
[
(Ar + Ab1b1,i+1 + Ab2b2,j +Br1K1 +Br2K2) 0
01×7 0
]
Aijpm =
[
(Ar + Ab1b1,i + Ab2b2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
, Aijmm =
[
(Ar + Ab1b1,i+1 + Ab2b2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
(5.37)
When i = 1, . . . , N − 1 and j = N + 1, . . . , 2N − 1;
Aijpp =
[
(Ar + Ab1a1,i+1 + Ab2b2,j +Br1K1 +Br2K2) 0
01×7 0
]
, Aijmp =
[
(Ar + Ab1a1,i + Ab2b2,j +Br1K1 +Br2K2) 0
01×7 0
]
Aijpm =
[
(Ar + Ab1a1,i+1 + Ab2b2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
, Aijmm =
[
(Ar + Ab1a1,i + Ab2b2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
(5.38)
finally, if i = N + 1, 2N − 1, j = 1, . . . , N , define
Aijpp =
[
(Ar + Ab1b1,i + Ab2a2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
, Aijmp =
[
(Ar + Ab1b1,i+1 + Ab2a2,j+1 +Br1K1 +Br2K2) 0
01×7 0
]
Aijpm =
[
(Ar + Ab1b1,i + Ab2a2,j +Br1K1 +Br2K2) 0
01×7 0
]
, Aijmm =
[
(Ar + Ab1b1,i+1 + Ab2a2,j +Br1K1 +Br2K2) 0
01×7 0
]
.
(5.39)
When i = N and j = 1, . . . , N − 1 u1 hits its positive saturation limits and the statespace matrices defining the LDI become
ANjmm = ANjpm =
[
Ar + Ab1a1,N + Ab2a2,j +Br2K2 Br1a1,N
01×N 0
]
ANjmp = ANjpp =
[
Ar + Ab1a1,N + Ab2a2,j+1 +Br2K2 Br1a1,N
01×N 0
](5.40)
while if j = N , i = 1, . . . , N − 1, u2 reaches its upper bound and
AiNmm = ANjmp =
[
Ar + Ab1a1,i + Ab2a2,N +Br1K1 Br2a2,N
01×N 0
]
AiNpm = ANjpp =
[
Ar + Ab1a1,i+1 + Ab2a2,N +Br1K1 Br2a2,N
01×N 0
]
.
(5.41)
similar considerations, omitted for brevity, can be made to describe the system behavior
inside the cells corresponding to input saturation. In general, a polytopic LDI in the form
d
dt
[
z
1
]
= co Aijmm, Aijpm, Aijmp, Aijpp[
z
1
]
(5.42)
can be adopted to describe the system behavior inside the cell Ωij , obtaining a piecewise
differential inclusion representation of the constrained bilinear closed-loop system over the
state space region Ic.
114
5.4. Stability and tracking domain analysis via piecewise quadratic Lyapunov functions
5.4.2 Piecewise quadratic Lyapunov function
The piecewise LDI description introduced in the previous Subsection, can then be analyzed
with non quadratic stability tools, in order to further reduce conservatism. In particular
a continuous piecewise quadratic Lyapunov candidate V (z) is considered, applying the
results presented in 4.4.1 to the power converters, in a similar fashion as what proposed
in ([107]) for a single input converter.
Based on the specific partition of Ic by (2N + 1)2 four vertices polytopes (see Fig. 5.3),
the following matrices can be introduced to ensure continuity of V (z) along the cells
115
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
boundaries
F0 =
[
In
04N×n
]
, Fi0 =
In 0
K1 −a1,1...
.
..
K1 −a1,i
03N−i×n 0
F0j =
In 0
02N×n 0
K2 −a2,j...
.
.
.
K2 −a2,j
02N−j×n 0
, Fij =
In 0
K1 −a1,1...
.
.
.
K1 −a1,i
02N−i×n 0
K2 −a2,1...
..
.
K2 −a2,j
02N−j×n 0
for i, j = 1, . . . , N
Fi0 =
In 0
0N×n 0
K1 −b1,1...
.
..
K1 −b1,i
03N−i×n 0
, F0j =
In 0
03N×n 0
K2 −b2,1...
.
..
K2 −b2,j
0N−j×n 0
Fij =
In 0
K1 −a1,1...
.
.
.
K1 −a1,i
02N−i×n 0
0N 0
K2 −b2,1...
.
.
.
K2 −b2,j
0N−j 0
for i = 1, . . . , N, j = N + 1, . . . , 2N
Fij =
In 0
0N×n 0K1 −b1,1...
..
.
K1 −b1,i
0N−i×n 0
K2 −a2,1...
.
.
.
K2 −a2,j
02N−j 0
for i = N + 1, . . . , 2N, j = 1, . . . , N
Fij =
In 0
0N×n 0K1 −b1,1...
.
.
.
K1 −b1,i
02N−i×n 0
K2 −b2,1...
.
..
K2 −b2,j
0N−j 0
for i = N + 1, . . . , 2N, j = N + 1, . . . , 2N.
(5.43)
116
5.4. Stability and tracking domain analysis via piecewise quadratic Lyapunov functions
Then, the piecewise quadratic Lyapunov candidate function can be defined as
V (z) =
zTF T0 PF0 z ∈ Ω0
[z 1]TF TijPFij
z
1
z ∈ Ωj
. (5.44)
with P ∈ R2N+7×2N+7 a symmetric positive definite matrix to be defined.
5.4.3 Invariance and set inclusion LMI conditions
Now the stability analysis for a given feedback matrix K can be extended relying on
function (5.44). As usual, the unit level set of V LV (1) := z : V (z) ≤ 1 is used as
an estimate of the stability region, and, provided its invariance (V < 0∀z ∈ LV (1)),
maximized w.r.t a given shape reference set. Since the piecewise LDI representation holds
only inside the region Ic, the inclusion condition LV (1) ⊂ Ic has to be fulfilled.
First a sufficient invariance condition is derived. To this aim, note that each polytopic cell
can be characterized by means of two quadratic inequalities: as regards Ω0 it holds
Ω00 =
z :
∣∣∣∣K1z −
a1,1 + b1,12
∣∣∣∣
2
≤(a1,1 − b1,1
2
)2
,
∣∣∣∣K2z −
a2,1 + b2,12
∣∣∣∣
2
≤(a2,1 − b2,1
2
)2
(5.45)
or, equivalently: zTM0iz ≤ 0, i = 1, 2 with z = [z 1]T and
Mi00 =
[
2KTi Ki −(ai,1 + bi,1)K
Ti
−(ai,1 + bi,1)Ki 2ai,1bi,1
]
≤ 0.
the same reasoning can be applied to the other cells. Since Ω0 include the origin in its
interior, the invariance condition V (z) > 0, V < 0 along (5.34) can be expressed as
F T0 PF0 > 0
AT0ppF
T0 PF0 + (AT
0ppFT0 PF0)
T < 0, AT0pmF
T0 PF0 + (AT
0pmFT0 PF0)
T < 0
AT0mpF
T0 PF0 + (AT
0mpFT0 PF0)
T < 0, AT0mmF
T0 PF0 + (AT
0mmFT0 PF0)
T < 0.
(5.46)
While for a generic cell not containing the origin, by S-procedure the invariance condition
along (5.42) can be expressed as
F TijPFij + α1ijM1ij + α2ijM2ij > 0
ATijppF
TijPFij + (AT
ijppFTijPFij)
T − β1ijM1ij − β2ijM2ij < 0
ATijpmF
TijPFij + (AT
ijpmFTijPFij)
T − γ1ijM1ij − γ2ijM2ij < 0
ATijmpF
TijPFij + (AT
ijmpFTijPFij)
T − δ1ijM1ij − δ2ijM2ij < 0
ATijmmF
TijPFij + (AT
ijmmFTijPFij)
T − η1ijM1ij − η2ijM2ij < 0, i, j = 1, . . . (2N + 1)
(5.47)
for non negative scalars β1ij , β2ij, γ1ij , γ2ij, η1ij , η2ij.
As concerns the set inclusion requirement LV (1) ⊆ Ic, it is easy to verify that it is satisfied
if and only if V (z) > 1 for all z belonging to one of the four hyperplanes K1z = ±c,
117
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
K2z = ±c, i.e. if V (z) − K1z/c > 0 for K1z = c, V (z) + K1z/c > 0 for K1z = −c,V (z) − K2z/c > 0 for K2z = c, V (z) + K2z/c > 0 for K2z = −c. According to the
previously defined partition, there are (2N + 1) × 4 cells sharing at least one boundary
with these hyperplanes. Thus, by S-procedure we can claim that the set inclusion condition
holds if there exist four groups of (2N + 1) scalars λNj , λ2Nj , λiN , λi2N , and four groups
of (2N + 1) non-negative scalars χNj , χ2Nj , χiN , χi2N , with i, j = 0, . . . , 2N such that
F TNj1PFNj −
1
2c
[
0 KT1
K1 0
]
+ λNj
[
0 KT1
K1 −2c
]
+ χNj+M2Nj > 0
F T2NjPF2Nj +
1
2c
[
0 KT1
K1 0
]
+ λ2Nj
[
0 KT1
K1 2c
]
+ χ2NjM2Nj > 0
F TiNPFiN − 1
2c
[
0 KT2
K2 0
]
+ λiN
[
0 KT2
K2 −2c
]
+ χiNM1iN > 0
F Ti2NPFi2N +
1
2c
[
0 KT2
K2 0
]
+ λi2N
[
0 KT2
K2 2c
]
+ χi2NM1i2N > 0
(5.48)
where the matrices F and M are associated with the (2N + 1) × N cells that shares a
boundary with the hyperplanes defining Ic.
Finally the size of LV (1) has to be measured w.r.t. some shape reference set XR in order to
define the maximization objective: ǫ∗ = sup ǫ : ǫXR ⊂ LV (1). The inclusion condition
is equivalent to require V (z) ≤ 1 for all z belonging to the boundary of the ball, i.e.
V (z) ≤ zTR
ǫ2z, ∀z ∈ ∂(ǫXR) (5.49)
where, for the sake of simplicity the reference set XR has been assumed to have an ellip-
soidal form. The above condition must be checked in each set Ωij , and, by S-procedure,
it is easy to verify that it is implied by the following matrix inequalities
[
F T0 PF0 0
0 0
]
− ω100M100 − ω200M200 + ζ0
[
0 0
0 1
]
≤ t
[
1 + ζ0R 0
0 0
]
F TijPFij − ω1ijM1ij − ω2ijM2ij + ζj
[
0 0
0 1
]
≤ t
[
1 + ζijR 0
0 0
]
, i, j = 1, . . . , 2N + 1.
(5.50)
with t = 1/ǫ2, ζ0, ζij ≥ 0.
5.4.4 Stability region estimation via LMI optimization
In view of all the considerations made in the previous Subsections, the stability region of
the origin of system (5.32) (corresponding to a generic working point xe of (5.23)), can
be determined by means of the Lyapunov candidate (5.44) by solving the optimization
problem (a GEVP in t, P )
inf t, s.t. (5.47), (5.48), (5.50). (5.51)
118
5.5. Simulation and experimental results
Thus the results obtained in 5.3.3, on stability of the error system (5.30), can be improved
taking xe = 0 and the scaled unit ball as shape reference set, then compute (5.51) for the
matrices K given in (5.29). Choosing N = 3 and aj1 = upj/4, aj2 = upj/2, bj1 = −umj/4,
bj2 = −umj/2, j = 1, 2, c = 1 yields t = 0.0016 for the gains K5, and t = 0.013 for the
gains K25 corresponding to the optimal scale values ǫ∗ = 25.13, ǫ∗ = 8.78.
Hence the conservatism introduced by the single LDI description and quadratic stability
tools has been significantly reduced, in particular the result obtained for η = 5 ensure
a practically global stability of the system as compared to realistic range of the circuit
variables. If the gains are increased to improve performances then the basin of attraction
is shrunk, however, the piecewise LDI description, and thus the obtained results, can be
enhanced by considering a more resolute state space partition, i.e. increasing N .
5.5 Simulation and experimental results
In order to test the effectiveness of the proposed control solution a detailed circuit simu-
lation, taking into account the switching nature of the power converters has been carried
out by means of MATLAB SimPower Systems toolbox. Then an experimental system
was constructed for the hybrid energy storage configuration as described in Fig. 5.1. As
mentioned in 5.3.3, the considered circuit is characterized by the parameters reported in
Tab. 5.2. The battery is a lead-acid one rated 6V, 13Ah, with an open circuit voltage
variation range between 5.8V and 6.3V, corresponding to 10% and 100% state of charge,
respectively, while the supercapacitor consist of two parallel ones rated 58F, 16.2V with
serial resistance 0.022Ω. Simulations with both the full order averaged model (5.14) and
the SimPower Systems circuit model have been carried out. The initial conditions for the
battery and the supercapacitor voltages have been set to E0 = 6V , Eu0 = 7V respectively.
Fig. 5.4 shows the simulation results carried out under an abrupt load resistance switch-
ing between the considered nominal value R0 = 2Ω (corresponding to a heavy load) and
R = 200Ω (simulating a light load). The adopted feedback law is defined by K25 in (5.29).
The references for the battery current ib and the load voltage vo are first set at 3A and
10V , as in the nominal condition; the obtained results are reported in Fig. 5.4(a). In each
plot, the responses by the averaged model are the smooth red curves and the responses
by the SimPowerSystem model are the blue curves. The load is switched 3 times in 1
second but it is long enough to reach a steady-state after each switch. From 0 to 0.25
second and from 0.5 to 0.75 second, the load resistance is 200Ω. During such light load,
both outputs (battery current and load voltage) converge to the set reference values, while
the supercapacitor current is negative. This shows that the supercapacitor is charged un-
der light load. From 0.25 to 0.5 second and from 0.75 to 1 second, the load resistance
is 2Ω. During such a heavy load, the outputs also converge to the set reference values
after some initial overshoot/undershoot. As a result, even if not a controlled output, the
supercapacitor supplies an average current greater than 5A so that the load power request
is satisfied. Recall that the feedback laws are designed for the nominal working condition
119
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
0 0.25 0.5 0.75 18
10
12
time [s]
v o[V
]
0 0.25 0.5 0.75 11
2
3
4
time [s]
i b[A
]
0 0.25 0.5 0.75 1−5
0
5
10
time [s]
i sc[A
]
(a) Response to load switch between 2Ω and 200Ω, ib,ref = 3A, vo,ref = 10V .
0 0.25 0.5 0.75 18
10
12
time [s]
v o[V
]
0 0.25 0.5 0.75 10
1
2
3
time [s]
i b[A
]
0 0.25 0.5 0.75 1−5
0
5
10
time [s]
i sc[A
]
(b) Response to load switch between 2Ω and 200Ω, ib,ref = 1.5A, vo,ref = 10V .
Figure 5.4: Simulation results under abrupt load switching.
where R0 = 2Ω. Hence the simulation results in Fig.5.4(a) demonstrate the robustness of
the control solution against parameter changes. This feature can be formally confirmed
by applying the stability analysis in 5.4 to the resulting working point xe for R = 200Ω.
The robust tracking performance is also validated by varying the reference for the battery
current. Fig. 5.4(b) shows the tracking of references ib,ref = 1.5A, vo,ref = 10V . It can be
noted that both the battery current and the load voltage track their respective reference
while, in response to a smaller battery current, the supercapacitor current is adjusted
automatically. During light load, it absorbs smaller current as compared to the value in
Fig. 5.4(a); during heavy load, it supplies larger current.
The experimental tests were carried out on a setup equipped with a Panasonic LC-
R067R2P battery model and two supercapacitor modules of BMOD0058 MAXWELL 58
Farads-16.2 V DC. Same type of MOSFETs (FQP50N06L) were used for the two bidirec-
tional buck-boost converters, with switching frequency at 37kHz. As regards the control
law implementation, observing that the elements in the first 5 columns of matrices K5,
K25 in (5.29) are much smaller than those in the last two columns, the simplified feedback
law has been considered
u =
[
−1.16∫(ib − ib,ref )dt− 0.57
∫(vo − vo,ref )dt
1.8∫(ib − ib,ref )dt− 5.08
∫(vo − vo,ref )dt
]
(5.52)
hence, the two duty cylces D1 = u1 +D10 and D2 = u2 +D20 can be easily computed. In
fact, the above control law can be implemented with operational amplifiers. For the real
circuit, the initial battery voltage was about 6.15V, and the initial supercapacitor voltage
was between 7V and 7.4V , while the load was switched between 2Ω and 200Ω as for the
simulation tests. Also in this case the system robust tracking capability, under the law
(5.52), were tested setting the reference for vo at vo,ref = 10V while adopting two values
ib,ref = 1.5A, 3A as battery current reference.
120
5.5. Simulation and experimental results
(a) Tracking response to load switch,
ib,ref = 3A, vo,ref = 10V , R =
2Ω, 200Ω.
(b) Tracking response to load switch,
ib,ref = 1.5A, vo,ref = 10V , R =
2Ω, 200Ω.
Figure 5.5: Experimental results under abrupt load switching.
Fig. 5.5(a) shows the response when the load was switched between 200Ω and 2Ω, with
ib,ref = 3A, vo,ref = 10V . The set up was similar to that for the simulation in Fig.
5.4(a) but the load switch was implemented manually and the total time was 6 seconds
(the time scale is 500ms/div). The four curves from top to bottom are respectively,
the supercapacitor current isc (Channel 4, orange, 5A/div), the supercapacitor’s terminal
voltage v1 (Channel 3, green, 5V/div), the battery current ib (Channel 2, red, 2A/div), and
the load voltage vo (Channel 1, blue, 10V/div). Initially the load was set at 200Ω. The first
1.5 seconds show a “steady state” (ignoring the slow increase of supercapacitor voltage)
for this load condition. The references for both outputs are tracked: ib = 3A, vo = 10V ,
except for measurement noises. The supercapacitor voltage is about 7V and its current
about -1.5A. At about 1.5 second, the load is switched to 2Ω. After a brief transient,
both the load voltage and the battery current quickly returned to the respective reference
value. This verifies the robust tracking performance under varying load. Meanwhile, the
supercapacitor current quickly increases to above 5A and continue to increase at a slower
rate. Since the heavy load condition last for about 2.7 seconds, much longer than the 0.25
seconds in simulation (in Fig. 5.4(a)), a slight decrease of supercapacitor voltage can be
noticed. Thus, to maintain a desired output voltage and power, the supercapacitor current
must increase. At about 4.2 second, the load is switched back to 200Ω. Again, after some
transient response, both outputs return to the desired reference values. Note that the
outputs settling time last more than 0.4 seconds for the experimental test; the reason is
that a simple model for the battery was used in simulation, if it was refined adding some
pairs of parallel resistor and capacitor, the transient response would last longer even in
simulation.
Fig. 5.5(b) shows the experimental results for the second reference scenario: where ib,ref =
1.5A, vo,ref = 10V . Also in this case the circuit behavior is similar to what observed in
simulation in Fig. 5.4(b). The assignment of the four channels are the same as in Fig.
5.5(a). The scale for Channel 4 is 5.5A/div as compared with 5A/div in Fig. 5.5(a).
The scales for other channels are the same. The load is switched from 200Ω to 2Ω at
121
Chapter 5. CONTROL DESIGN FOR POWER CONVERTERS FED BY HYBRID ENERGY SOURCES
about 1.3 second, then switched back to 200Ω at about 4.07 second. Both outputs follow
their references after each load switch. As expected, the supercapacitor drew less current
during light load period and supplied more current during heavy load period. This test
also verifies the robust tracking performance under different reference values for a realistic
circuit.
122
Chapter 6
Saturated Speed Control of
Medium Power Wind Turbines
In this chapter a specific saturated control solution, regarding the speed control
of wind turbines is presented. The approach to handle control input saturation is
based on a suitable partition of the control effort between the speed control knobs,
i.e. blade pitch angles and generator torque, avoiding hybrid control structures
which are impaired by wind and turbine dynamics uncertainties. Time-varying
torque and power limits at generator side, related to the system thermal dynamics
are taken into account, to better exploit the generator capabilities. A full stability
analysis under unknown wind speed and uncertain aerodynamics curves is carried
out, showing how to tune the proposed controller for a wide stability domain.
Furthermore a standard MPPT algorithm is mounted on top of the proposed
solution.
6.1 Introduction and motivation
Wind energy conversion has been growing extremely fast along the last decade, becoming
the most competitive energy source among the renewable sources for electrical power gen-
eration. Thanks to the improvement in wind turbine and power electronics technologies,
today variable-speed pitch-regulated wind turbines ([111]) are usually adopted in medium
or large scale power production, maximizing the energy captured from the wind in almost
every working condition.
Basically, two main kinds of variable-speed, pitch-regulated wind turbines can be iden-
tified: large power ones (a few MW) and medium power ones (a few hundreds of kW).
These categories show, at the same time, similarities and differences, affecting the con-
trol requirements. Both kinds of turbines are expected to capture the maximum power
available from the wind, up to the rated power of the electric generator drive. Hence, two
main working region can be defined; “at low wind speed”, with an available wind power
lower than the nominal turbine power, and “at high wind speed” where available wind
123
Chapter 6. SATURATED SPEED CONTROL OF MEDIUM POWER WIND TURBINES
power is equal or greater than the turbine nominal one. At low wind speed, the generator
torque, the turbine speed and the pitch angles need to be regulated in order to capture
the maximum power from the wind according to turbine aerodynamics and wind speed.
Differently, at high wind speed, pitch angles and generator torque are set to bound the
extracted power to the nominal one and, often, to keep a constant turbine speed. Beside
this common general strategy, large power and medium power turbines show relevant dif-
ferences in two main elements: concerns about mechanical vibrations and knowledge of
accurate mechanical and aerodynamic models.
For large power turbines, torsional vibrations of the drive line and tower and blades oscil-
latory modes need to be carefully damped, since relevant fatigue stress can occur, owing to
large mechanical loads and large dimensions (natural mode frequencies are rather low with
respect to the operating frequency range). On the other hand, for this kind of turbines,
accurate aerodynamic and mechanical models are available together with wind speed sen-
sors. The large cost of such plants motivates the strong effort in modeling and sensoring.
These features enable to define an offline optimizing curve for “low wind speed” region
and to design accurate multivariable control algorithms to deal with mechanical vibra-
tions. Various solutions have been presented in literature; the most popular methodology
is based on linearization along trajectories or equilibria and application of advanced LPV
gain scheduling approaches, exploiting H2, H∞ techniques to shape performances and ro-
bustness to some model uncertainties ( [112], [113], [114]). Differently, in [115] the turbine
control problem has been cast into a receding horizon nonlinear adaptive model predictive
control framework in order to enhance performance under off-design conditions.
A different scenario takes place when medium power turbines are considered. Mechanical
vibrations are no longer a crucial issue (natural modes are usually outside the working fre-
quency range, thanks to smaller dimensions, and mechanical loads are not critical), on the
other hand, wide dispersion of aerodynamic characteristics usually affects these turbines
and very poor models and measurements are available, owing to development and pro-
duction cost limitation. Therefore, Maximum Power Point Tracking (MPPT) algorithms
have received particular attention to steer adaptively the turbine toward the best working
condition in the “low wind speed” region. MMPT solutions are usually structured as hill-
climbing discrete-events searching algorithms. In [116], [117] and [118], the turbine speed
is modified to search for the power optimum, while in [119] the convexity of the parabola
representing the generator torque curve is adapted. Whatever solution is used, a crucial
issue is to guarantee stability of the wind turbine in any condition, taking into account the
electric power saturation occurring in “high wind speed” region. In [119], where generator
torque is modified for MPPT, a stability analysis is presented only for low wind speed
operation and without considering pitch control. Differently, in approaches where turbine
speed is varied for MPPT (the most common ones), the focus is on the algorithm efficiency
and no formal stability analysis is usually carried out, implicitly assuming that a robust
closed-loop speed control is present.
In this chapter, a simple and effective speed controller, first introduced in [39] is presented.
124
6.2. System Modeling and Control Objective
Differently from the applications considered so far, the saturated speed controller does not
rely on the theoretical backgrounds presented in chapters 1 and 4, however a sort of one
step approach is adopted exploiting a suitable partitioning of the overall control effort
between the two system “knobs”; i.e. blade pitch angles and generator torque. It will be
showed how this solution does not require a switching between two different controllers cor-
responding to the previously defined operating regions, this allows to intrinsically prevent
from possible bumps and limit cycles under variable wind and uncertain aerodynamics
characteristics. A basic PI structure is proposed and a complete stability analysis un-
der unknown wind speed and uncertain aerodynamics curves is carried out defining some
tuning rules for for a wide stability domain under time varying torque/power saturation
limits.
The chapter is organized as follows. In Section 6.2 the medium-power wind turbine model
is reported and the general saturated control strategy is discussed, in Section 6.3 the pro-
posed speed control solution is presented, and its stability properties and the related design
rules are discussed. In Section 6.4 a slight modification of a common MPPT algorithm is
presented in order to be “mounted on top” of the proposed speed controller. Simulation
results are presented in Section 6.5.
6.2 System Modeling and Control Objective
The class of wind turbines considered in this chapter are medium-power, horizontal axis,
variable-speed variable-pitch wind turbine, with collective blade pitch actuation (according
to pitch-to-feather strategy, see[38]). Their rotational dynamics can be modeled as follows:
ω =1
J(Tw(c, V, ω, β)− TG) (6.1)
where ω, V are respectively the rotor and wind speed, J is the total rotational inertia,
collecting the blades, drive train shaft, and electric generator rotor contributions, and
ρ the air density, while TG is the actuated generator torque and Tw is the aerodynamic
torque which, assuming a perfect alignment with wind direction, can be expressed as ([38])
Tw(c, ω, β) =1
2ρπR3Cq(c, λ, β)V
2, Cq(c, λ, β) =Cp(c, λ, β)
λ. (6.2)
Thus, the power captured from the turbine can be derived from (6.2)
P =1
2Cp(c, λ, β)ρπR
2V 3, λ =ωR
V(6.3)
Cq, Cp are the so-called torque and power coefficient that define the aerodynamic of the
turbine, they depend on the blades pitch angle β, and the tip speed ratio λ, while the
vector c contains the coefficients of the function adopted to fit the turbine power curve.
Here the following approximation, valid for a wide range of commercial turbines ([120]),
is considered
Cp(c, λ, β) = c1
(c2λi
− c3β − ce
)
e
(
− c5λi
)
+ c6λ,1
λi=
1
λ+ 0.08β− 0.035
β3 + 1. (6.4)
125
Chapter 6. SATURATED SPEED CONTROL OF MEDIUM POWER WIND TURBINES
In order to take into account the approximation error with respect to the unknown actual
turbine curve, a limited set C such that c ∈ C, centered on the nominal values n-ple
c1n, . . . , c6n is considered for the vector c parameters.
In most of medium scale turbines the only accurately measured variables are the shaft
angular speed and the generator torque, hence also the generator power can be derived,
while accurate knowledge about inertial terms is usually available by manufacturers data.
On the other hand wind speed value is usually not available or roughly provided by an
anemometer whose measure usually does not fit accurately the actual wind field acting on
the blades, hence it cannot be used to obtain information about wind aerodynamic torque,
furthermore the uncertainty on Cp curve cannot be neglected.
It’s further to remark that blades flapping, tower fore-aft motion and drive train shaft
resonant modes will be neglected for control purposes, hence model (6.1) has been derived
assuming a perfectly rigid system, putting the focus on the main rotating dynamic of the
drive train shaft. This approximation can be effectively adopted to define a control law
for the class of turbines here considered, indeed in medium power wind energy conversion
systems often no gearbox is present and, as mentioned, resonant modes are at much higher
frequency and more damped than those in large power turbine and, then, usually outside
the control bandwidth.
6.2.1 Considerations on the general control strategy
From the expression of the aerodynamic power (6.3) it can be seen that the energy captured
from the wind can be varied shaping the power coefficient by means of the tip speed ratio
and the pitch angle; the maximum power coefficient corresponds to optimal values for
tip speed ratio and pitch angle λ∗, β∗. While λ∗ slightly depends on the system specific
aerodynamic characteristic, the pitch angle value maximizing the power coefficient, for all
kind of wind turbine, regardless power curve uncertainties, is β∗ = 0 ([38]).
Bearing in mind these considerations, in order to achieve maximum power extraction at
below rated wind speed, the pitch angle can be held constant to zero, while the angular
speed is varied to reach the optimal tip speed ratio; if information about the turbine
aerodynamic are available with high precision, the generator torque can be actuated as
a feedforward action following the optimal power extraction locus reported in Fig. 6.1,
with this method the equilibrium point corresponding to the optimal angular speed results
asymptotically stable. At high wind speeds, the angular speed increases until power-torque
saturation occurs, then the pitch angle is varied to shed aerodynamic torque and control
the angular speed.
An analogous strategy can be adopted without exploiting turbine aerodynamic and wind
speed knowledge, in this case the optimal tip speed ratio is reached by means of an MPPT
algorithm properly integrated with a speed controller, however, at high wind speed, the
strategy is different from the previous one. Controlling the angular speed, power-torque
saturation limit can be reached also at low rotor speed if a strong wind gust occurs,
in this case the MPPT algorithm is suspended and the pitch angle is used to shed the
126
6.2. System Modeling and Control Objective
0 5 10 150
1
2
3
4
5
6x 105
P [W
]wind speed [m/s]
Figure 6.1: Optimum power generation locus.
aerodynamic torque and ensure a constant power operation (highlighted in bold in Fig. 6.1)
at the maximum generator value. In Section 6.4 a slightly different strategy is proposed
to limit generator power losses due to high torque demand. Whatever control strategy
is adopted, the critical issue is to ensure a smooth transition between the two different
turbine operating regimes, providing reliable performance and stability for all the possible
conditions.
6.2.2 Torque and Power saturation
Before detailing the saturated control solution, the system torque and power saturation
need to be elaborated, accounting for the electrical machine and electrical power thermal
behavior. Typically the following input constraints have to be respected: maximum torque
and power peak values, TGmax and PGmax, and maximum RMS values, TGRMSmax and
PGRMSmax (obviously lower than the corresponding peak bounds). According to common
sizing rules, torque constraints (both peak and RMS) are related to the electrical machine
used as generator, while power constraints are related both to the electrical machine and
the controlled power converter, but mainly to the second one (which is used to drive the
generator and transfer the electric power to the line grid). In addition, RMS constraints are
actually related to thermal bounds on electric machine and power converter, then thermal
dynamics should be taken into account for a better exploitation of the actuators capabil-
ities. Usually this possibility is not considered and constant instantaneous limits equal to
TGRMSmax and PGRMSmax are adopted as generator torque and power constraints.
Here, thermal dynamics will be explicitly considered, in a receding horizon model pre-
dictive fashion, to derive temporary higher bounds than TGRMSmax and PGRMSmax (but
obviously lower than instantaneous peak limits and with some safety margin). Beside
a better exploitation of the generator and power electronics, these time-varying bounds,
combined with the control solution presented in Section 6.3, will allow a smoother pitch
angle variation.
The thermal behavior of both generator power drive and front-end converter can be ap-
proximated by a first order dynamic, hence the torque and power RMS constraints can be
127
Chapter 6. SATURATED SPEED CONTROL OF MEDIUM POWER WIND TURBINES
written as follows (with some abuse of notation between Laplace and time domain):
T 2GRMS(t) ,
T 2G(t)
1 + τ1s≤ TGRMSmax P 2
GRMS(t) ,P 2G(t)
1 + τ2s≤ PGRMSmax (6.5)
with P 2G = (TGω)
2 and where, according to common sizing, τ1 is the electric generator
thermal time constant (tipically a few minutes), while τ2 is the thermal time constant of
the converter grid-side (usually a few tens of seconds). The time-varying bound for the
generator torque, which can be applied without overcoming the RMS limitation can be
calculated as follows. A time prediction horizon T , reasonably shorter than the thermal
settling-time (around 5τ1), is selected. Then, inverting the model (6.5), the constant
torque value TG(t) that, applied to the system over the time horizon [t, t+T ], leads to the
limit value T 2GRMSmax starting from the initial value T 2
GRMS(t), is derived. The obtained
equation of this new time-varying receding horizon thermal bound, T , reported in the
following, enlighten that the bound will be always greater or equal than TGRMSmax
T (t) =
√√√√T 2
GRMSmax − T 2GRMS(t)e
− Tτ1
1− e− T
τ1
≥ TGRMSmax (6.6)
The same approach can be followed to calculate the power limit P similarly defined to T
P =
√√√√P 2
GRMSmax − P 2GRMS(0)e
− Tτ2
1− e− T
τ2
≥ PGRMSmax. (6.7)
Adopting these constraints, evaluated at run time by means of the described procedure,
in place of the steady state values TGRMSmax, PGRMSmax, the following torque saturation
law holds (power saturation law can be derived accordingly)
TGsat = min
(
TGmax,PGmax
ω, T ,
P
ω
)
. (6.8)
This saturation threshold is less conservative with respect to the usually adopted steady-
state values, the bounds that would be achieved without thermal dynamic consideration
are a lower bound of TGsat in (6.8) and can be expressed as follows
TGsatmin(ω) = min
(
TGRMSmax,PGRMSmax
ω
)
(6.9)
This bound will be useful for offline dimensioning and MPPT algorithms.
6.3 Saturated Speed Controller Design
The basic idea of the proposed solution is to consider a unique scalar control input for
speed regulation, given by the sum of the generator torque and the torque effect of the pitch
angle variation w.r.t. its optimum value for power capture (β = 0). Hence, a unique SISO
speed controller is designed for the whole operating range and its total torque command is
split in generator torque command and pitch angle command, so that the generator torque
128
6.3. Saturated Speed Controller Design
and power limits are never overcome. According to the general control strategy reported
in 6.2.1, when the total torque command is below the generator bounds, only a generator
torque command will be issued, while pitch angle is left at its optimum value. Differently,
when total torque command exceeds the generator bounds a suitable pitch angle variation
will be requested. This approach induces an intrinsically smooth transition between “low
wind speed” and “high wind speed” condition even if the generator torque-power limits
are time-varying and fast changes in the wind regime occur.
6.3.1 Controller definition
First of all, the mechanical model (6.1) is rearranged in order to separate the aerodynamic
torque generated with fixed optimum pitch angle, β = 0, from the braking effect obtained
by moving β to positive values. Hence, defining T∆ as:
T∆(c, V, ω, β) = Tw(c, V, ω, 0)− Tw(c, V, ω, β)
model (6.1) can be rewritten as
ω =1
J(Tw(c, V, ω, 0)− (T∆(c, V, ω, β) + TG)) (6.10)
where (Tw(V, ω, 0) (non-negative, by turbine physics) can be seen as an exogenous input
depending on wind, while the sum (T∆(c, V, ω, β) + TG) can be intended as a single scalar
control input (also T∆ is non-negative according to turbine physics). The distribution of
the total control input command in TG and β can be decided according to maximization
of the generated electric power and generator torque and power bounds. By the way, this
operation is completely independent of the chosen speed controller which “sees” a single
control command.
Before designing the speed controller with the above mentioned assumption on the control
input, it is necessary to note that the β contribution is actually dependent on the aero-
dynamic parameters c, and the wind speed, V which are uncertain and not measurable,
respectively. Hence, for control purposes, T∆(c, V, ω, β) is approximated by an averaging
procedure which yields the following function independent of parameters variation, wind
and turbine speeds
f(β) , mean T∆(c, V, ω, β) | c ∈ C, ω ∈ [0, ωmax], V ∈ [V ∗(c, ω), Vmax] (6.11)
the set C is the set of admissible aerodynamic parameters, ωmax is the maximum operating
speed allowed for the wind turbine, Vmax is the so-called survival wind speed (i.e. the max-
imum wind speed the turbine is designed to resist at) and V ∗(c, ω) is the wind speed which,
for given c and ω, generates TW (c, V ∗(c, ω), ω, 0) = TGsatmin. Hence, owing to monotonic-
ity of TW with respect to V , at given c and ω, V < V ∗ =⇒ TW (c, V, ω, 0) < TGsatmin,
while V ≥ V ∗ =⇒ TW (c, V, ω, 0) ≥ TGsatmin. The reason why the wind range in the
considered set is lower bounded by V ∗ is the following. According to the reasoning made
in 6.2.1, β will be used only when the total control command hit the available generator
129
Chapter 6. SATURATED SPEED CONTROL OF MEDIUM POWER WIND TURBINES
torque limit, this can occur, in principle, in any condition depending on current tracking
error and reference derivatives. However, restricting the analysis to most common condi-
tions, namely steady-state or almost steady-state (i.e. with null or small tracking error and
limited speed reference derivative), the use of β is expected to occur only when the wind
speed is high enough to generate an aerodynamic torque larger than the one available at
generator side. Hence it make sense to restrict the area used to calculate the mean value
of T∆ to the most common and significant one.
It is wort noting that f(0) = 0 by definition and, as clarified in the following, for control
purpose f(β) needs to be a bijective continuous function; therefore, a bijective continuous
approximation would be adopted if the mean value of T∆ was not. In principle a function
f(β, ω), instead of f(β), could be adopted to approximate T∆(c, V, ω, β) since ω is mea-
sured, however, even the simpler method discussed here ensures to obtained the desired
results. According to the definition of f(β), (6.10) can be rewritten as
ω = 1J (Tw(c, V, ω, 0)− [TG + f(β)]
︸ ︷︷ ︸−T∆(c, V, ω, β)
)
(6.12)
where, actually, the scalar control input will be the sum TG + f(β), denoted as u in
the following, while T∆ = T∆(c, V, ω, β) − f(β) represents the control input addictive
unknown error due to aerodynamic uncertainties (note that T∆(c, V, ω, 0) = 0, ∀c, V, ω,by definition).
After this model reformulation, enlightening the “combined” scalar control input, the
addictive error on control input and the unknown exogenous input (depending on wind
and turbine speed only), the speed controller can be designed. A simple PI structure
with feed-forward action is selected with the the main purpose of guaranteeing robust non
local stability properties, despite of the unknown disturbances and actuation errors. The
proposed controller is defined as follows:
u = kpω + χ− Jω∗
χ = kiω(6.13)
where ω = ω − ω∗ is the angular speed error with respect to the reference ω∗, which is
assumed to be known and bounded together with its time derivative ω∗. The integral
term χ has the basic role to estimate and compensate for the unknown exogenous input
Tw and the unknown actuation error T∆. Actually, these terms are constant, in perfect
tracking condition, only when ω∗, V and TGsat are constant. Then, perfect tracking ca-
pability is structurally limited to those conditions. Nevertheless, ω∗, V and TGsat, when
non-constant, are usually slowly varying, leading to small residual tracking errors..
For what concerns the distribution of the control command u on TG and f(β), accord-
ing to the general control strategy devoted to maximize the generated electrical power,
the following partitioning rule, explicitly accounting for the system saturation limits is
adopted:
(TG, β) =
(u, 0) ifu ≤ TGsat
(TGsat, f−1(u− TGsat)) ifu > TGsat
(6.14)
130
6.3. Saturated Speed Controller Design
Finally, it can be noted that, to avoid motoring behavior of the electric generator, negative
values of control input u should be prevented. This can be practically achieved by suitable
limitation on the speed reference derivative, assuming a bounded derivative in the wind
speed variation and starting with reasonably small speed tracking error. Nevertheless,
considering very large initial error, the positiveness and boundedness of u can be assured
by saturating it at [0, umax] (with umax depending on generator and pitch properties of the
turbine) and adding a suitable anti-windup strategy for the PI controller. More details
about this issue are reported in appendix B, where practical anti-windup solutions for
SISO PI controllers are briefly discussed.
6.3.2 Stability analysis
The stability analysis of the proposed solution, leading to rules for PI gains selection, is
carried out assuming constant wind speed V , turbine velocity reference ω∗ and generator
torque saturation TGsat. The proposed results can be extended to slowly varying conditions
for the above mentioned variables, by considering their derivatives as sufficiently small
perturbing inputs. The control objective is to asymptotically stabilize the system (6.12)
with controller (6.13) and the partitioning rule (6.14) at the following equilibrium point
ω = ω∗ i.e. ω = 0, χ = Tw(c, V, ω∗, 0)− T∆(c, V, ω
∗, β) (6.15)
with
β =
0 if Tw(c, V, ω∗, 0) ≤ TGsat
¯β s.t. Tw(V, ω∗, 0)− T∆(V, ω
∗, ¯β)− TGsat = f( ¯β) otherwise
Defining χ = χ− χ, the following error dynamics can be derived
˙ω =1
J
(
Tw(c, V, ω, 0)− T∆(c, V, ω, β)−Kpω − χ− χ)
˙χ = Kiω.(6.16)
The displacement of TW and T∆ with respect to their values at the equilibrium point can
be defined as follows
˜T∆(c, V, ω∗, ω, β) = T∆(c, V, ω, β)− T∆(c, V, ω
∗, β)
Tw(c, V, ω∗, ω) = Tw(c, V, ω, 0)− Tw(c, V, ω
∗, 0)(6.17)
hence, by (6.17) and (6.15), the error dynamics system (6.16) can be rewritten as
˙ω =1
J
(
Tw(c, V, ω∗, ω)− ˜T∆(c, V, ω
∗, ω, β)− kpω − χ)
˙χ = kiω.(6.18)
Then, it can be noted that the computation of β in (6.14) can be rewritten as
β = f−1 (sat∞0 (kpω + χ+ χ− TGsat)) . (6.19)
131
Chapter 6. SATURATED SPEED CONTROL OF MEDIUM POWER WIND TURBINES
Therefore the error dynamics can be rewritten enlightening the remarkable dependence of˜T∆ on kpω and χ, which is crucial for stability analysis and PI design,
˙ω =1
J
(
−kpω − χ+ Tw(c, V, ω∗, ω)− ˜T∆ (c, V, ω∗, ω, kpω, χ)
)
˙χ = kiω(6.20)
After the above rearrangements of the error dynamics, the results on stability analysis can
be summarized in the following proposition.
Proposition 6.3.1 Consider system (6.20),
- if there exist six positive numbers, k1, k2, k3, ωmax, ymax and χmax, such that
• the following inequality
|Tw(c, V, ω∗, ω)− T∆(c, V, ω∗, ω, y, χ)| < k1|ω|+ k2|y|+ k3|χ| (6.21)
holds uniformly w.r.t. c ∈ C, V ∈ [0, Vmax] and ω∗ ∈ [0, ωmax], and ∀ω ∈ [−ωmax, ωmax],
∀y ∈ [−ymax, ymax] and χ ∈ [−χmax, χmax],
• the following system of inequalities
kp (1− 2k2 − k3) > 2k1
k2p(1− k22 − 2k2 − 2k3)− 2kp(k2k1 + k1)− k21 > 0(6.22)
admits solution kp > 0,
- if kp > 0 is selected, independently of c, V and ω∗, but satisfying (6.22), and if ki
is chosen as ki = k2p/(2J), then the origin of (6.20) is asymptotically stable, for each
constant c, V and ω∗ in the domain where (6.21) holds, with a basin of attraction Ω =
(ω, χ) :kp8 ω
2 + 12
(kp2 ω + χ
)2< V ∗
where
V ∗ = sup
V : ∀(ω, χ)with kp8 ω
2 + 12(
kp2 ω + χ)2 = V it holds
ω ∈ [−minωmax, ymax/kp,minωmax, ymax/kp]and χ ∈ [−χmax, χmax]
Proof Setting ki = k2p/(2J) and introducing the following change of coordinates
z1 =kp2ω, z2 =
kp2ω + χ (6.23)
the system (6.20) can be expressed as
z1 = − kp2Jz1 −
kp2Jz2 +
kp2J
(
Tw − ˜T∆
)
z2 = − kp2Jz2 +
kp2Jz1 +
kp2J
(
Tw − ˜T∆
)
.
(6.24)
132
6.3. Saturated Speed Controller Design
Consider now the following candidate Lyapunov function V = 12(z
21 + z22) taking the
derivative along the system trajectories yields
V = − kp2J
(z21 + z22
)+kp2J
(Tw − ˜T∆)(z1 + z2). (6.25)
According to (6.21), the following inequality holds, for all c ∈ C, V ∈ [0, Vmax], ω∗ ∈
[0, ωmax], χ ∈ [−χmax, χmax] and ω ∈ [−min(ωmax, ymax/kp), min(ωmax, ymax/kp)]:
Tw − T∆ <k1|ω|+ k2|kpω|+ k3|χ| = k1
∣∣∣∣
2z1kp
∣∣∣∣+ k2|2z1|+ k3 |z2 − z1| <
k1
∣∣∣∣
2z1kp
∣∣∣∣+ k2|2z1|+ k3 |z1|+ k3 |z2|
hence, in the same domain, the following inequality for V holds
V < − kp2J
(z21 + z22) +kp2J
((
2kp
+2k2+k3)
|z1|+k3|z2|)
(|z1|+ |z2|)
the quadratic form on the right-hand side can be rewritten as
V < −[
|z1| |z2|][
kp2J − kp
(k2J + k3
2J
)
− k1J
−k1−kpk2−kpk32J
−k1−kpk2−kpk32J
kp(1−k3)2J
][
|z1||z2|
]
(6.26)
and, according to the assumption on kp > 0 satisfying (6.22), it will be negative definite.
6.3.3 Numerical results for a case of study
In order to show that the above proposed approach can lead to reasonable and feasible
controllers for practical medium power conversion energy system, a typical 200kW three-
blades wind turbine, with blade length of R = 13m, is considered as case of study. A
direct-drive coupling between turbine and electric generator is also assumed. The following
nominal coefficients for Cp curve expression (6.4), corresponding to the values reported in
MATLABTM wind turbine library, have been adopted: c1n = 0.517630, c2n = 116, c3n =
0.4, c4n = 5, c5n = 21, c6n = 0.0067950, and the set of variation C, defined considering a
spread of ±10% around these values, has been considered. In Fig.6.2 the power coefficient
surface obtained putting the nominal n-ple into expression (6.4) is reported. Numerical
values of all the system parameters, adopted to test the proposed solution, are summarized
in Tab. 6.1. Note that, owing to the front-end converter fast thermal dynamics (τ2 is
negligible w.r.t. other dynamics), the P for the considered system, calculated by (6.7), will
be always equal to PGRMSmax. Based on these parameters the approximation function
f(β) is derived applying the mean value calculation reported in (6.11); the obtained curve
is reported in Fig. 6.3. For what concerns controller gains, first, taking the error variables
bounds ωmax = 5 rad/s, ymax = 1×106Nm, χmax = 10×103Nm, the following parameters
are selected in order to obtain the conservative disturbances approximation expressed by
inequality (6.21); k1 = 4800, k2 = 0.23, k3 = 0.23, then plugging these values into the
system of inqualities (6.22), the stability condition on proportional gain: kp > 437663, is
derived. Ffinally the corresponding integral gain ki = 2.4× 106 can be set as indicated in
Prop. 6.3.1.
133
Chapter 6. SATURATED SPEED CONTROL OF MEDIUM POWER WIND TURBINES
0
5
10
15 0 5 10 15 20 25 30
−1.5
−1
−0.5
0
0.5
Cp
βλ
Figure 6.2: Power coefficient surface versus tip speed ratio λ and pitch angle β.
Wind turbine characteristics
System inertia J [Kgm2] 40000
Shaft speed range [ωmin, ωmax] [rad/s] [0,15]
Nominal angular speed ωnom [rad/s] 6.7
Wind speed range for power production [Vcin, Vcoff ] [m/s] [3.5, 25]
Rated wind speed Vnom [m/s] 11
Survival wind speed Vmax [m/s] 56
Maximum generator torque peak TGmax [Nm] 50000
Maximum generator power peak PGmax [kW] 350
Maximum RMS generator torque TGRMSmax [Nm] 30000
Maximum RMS converter power PGRMSmax [kW] 200
Generator thermal time constant τ1 [s] 60
Converter thermal time constant τ2 [s] negligible
Prediction time horizon T [s] 40
Pitch angle range [βmin, βmax] [deg.] [0, 60]
Table 6.1
6.4 Combination with MPPT approaches for speed refer-
ence generation
A standard MPPT algorithm can be easily mounted on the proposed control structure to
generate an optimal speed reference that allows to track the maximum power curve for
the turbine when needed, and to manage also the other operating conditions.
Overall control logic and speed reference generator, in general, has to manage four differ-
ent phases: system start up, maximum power tracking, power saturation and switch off.
Among these the most significant are MPPT and power saturation phases. For MPPT a
discrete event algorithm, based on a perturb and observe approach, is adopted following
these steps: starting from a constant speed ω∗(0), reached after startup, a first attempt
perturbation of the reference angular speed is produced with a predefined ∆ω∗(0), i.e.
134
6.4. Combination with MPPT approaches for speed reference generation
0 10 20 30 40 50 60−5
0
5
10x 10
5
pitch angle [deg]f
[Nm]
Figure 6.3: T∆ mean value over c, ω, V set of variation calculated for all pitch angle
admissible values.
ω∗ = ω∗(0) + ∆ω∗(0). As detailed in the following, the new speed reference is smoothly
applied to the controller, then, when steady-state condition is reached (i.e. constant speed
reference and small tracking error), mean value of the electric power captured by the gen-
erator is evaluated and assumed equal to the aerodynamic power P , extracted from the
turbine, see (6.3), possibly saturated by the generator torque bound at very low rotor speed
(this condition is actually rather unusual). The ratio between generator power variation
and the imposed speed reference variation is adopted as a local gradient estimation of the
P − ω curve. This estimate, scaled by a suitable coefficient K, is then used to define the
subsequent reference speed perturbation and to restart the procedure, until the estimated
gradient is sufficiently close to zero. The above mentioned method can be summarized
with the following equations to be recursively applied at each step k ≥ 1.
∆PG(k) = PG(k)− PG(k − 1) ∆ω∗(k) =∆PG(k)
∆ω∗(k − 1)
ω∗(k + 1) = ω∗(k) +K∆ω∗(k)
convergence of the proposed gradient based method to the global maximum of PG can be
ensured, assuming constant wind speed and pitch angle during the algorithm computation
steps, by concavity in the variable ω, of function (6.3) expressing the power extracted
from the wind (see also 6.4). In this respect, the scaling parameter K has to be properly
to achieve the desired convergence properties. Several methods are available, from exact
search methods optimizing PG along the ray ω∗(k)+K∆ω∗(k) to steepest increase methods
(see [91]). Here a so-called guarded Newton-Raphson method is considered by choosing K
such that
K <
(
max
∣∣∣∣
∂2P (c, V, ω, β)
∂2ω
∣∣∣∣
)−1
(6.27)
where the maximum value of partial second derivative is evaluated on the following set
(c, V, ω, β) : c ∈ C, V ∈ [0, Vmax], ω ∈ [0.7, 1.3] · λopt(c)V/R, β = 0.Once the almost maximum power condition is reached (sufficiently small gradient estima-
tion is obtained), the MMPT algorithms is stopped and restarted when a relevant variation
in the generated power occurs, since it is a symptom of a possible wind variation requiring
a new MPPT run.
135
Chapter 6. SATURATED SPEED CONTROL OF MEDIUM POWER WIND TURBINES
For what concerns the smooth application of the speed reference perturbation to the con-
troller, it is worth noting that this issue is relevant not only to reduce stress on the speed
controller, but also to avoid motoring behavior of the generator in transient condition (i.e.
to avoid that the speed controller requires a negative torque TG, drawing electric power
from the line grid). A filtered ramp is used for speed perturbation generation and its slope
is run-time adapted in order to take into account the torque available form the wind. In
particular, at each iteration k of the MPPT algorithm, the adopted reference slope ω∗(k)
is defined as follows
ω∗(k) =ηTG(k)
J(6.28)
where TG(k) is the mean value of the torque applied, with fixed speed reference, at k
iteration and represent an estimation of the torque available from the wind; η << 1,
typically 0.2, guarantees that only a small fraction of the torque available from the wind
is used to change turbine angular speed, avoiding an excessive torque demand that would
cause energy injection to accelerate the blades.
Transition from the MPPT phase to power saturation phase takes place when mean power
at generator side reaches or exceeds the value PGRMSmax at a certain iteration k; note that
this limit can be exceeded for a limited amount of time, owing to the thermal time-varying
bound adopted in speed control. This will occur when the wind available power is larger
than the turbine/generator nominal one. At this stage, the speed reference is smoothly
increased going over the nominal value ωnom that is the value at which, for rated wind
speed Vnom the captured power equals Pmax. This procedure allows to reduce the generator
torque value while keeping constant generator power by means of pitch angle, hence lower
currents need to be drained by the generator power drive and the thermal power losses
are minimized. It’s further to notice that the value of reference speed to impose when
power saturation occurs, has to be accurately selected, the rational is to choose a safe
angular speed value that ensures the turbine braking even if a wind gust up to survival
wind speed takes place. Finally, the reverse transition from power saturation phase back
to MPPT phase will occur whenever the mean generator power significantly fall below the
PGRMSmax bound.
6.5 Simulation results
Extensive simulations have been carried out to test the proposed solution. The tur-
bine characteristics reported in 6.3.3 have been considered, moreover, the following non-
idealities and bounds have been added to account for a realistic pitch actuator: a first
order dynamics between pitch command and actual pitch position with a time constant
τ = 20ms, a slew rate limitation at ±10/s and a limited pitch angle range, β ∈ [0, 60].
Finally, in order to prove the proposed solution effectiveness for actual wind energy conver-
sion systems, a discrete-time implementation of the controller has been carried out; taking
into account the common performances of turbine controllers, a sampling time Ts = 4ms
has been selected for the considered case of study.
136
6.5. Simulation results
0 100 200 300 4000
15
30
45Actuated pitch angle
0 100 200 300 40002468
10Reference angular speed
0 100 200 300 4000
1.5
3.55
x 104 Generator torque and bound (red)
0 100 200 300 400−0.2−0.1
00.10.2
Angular speed error
0 100 200 300 4000
15
30
45Mean wind speed
V[m
/s]
ω∗
[rad/s]
ω[rad/s]
TG,TG
sat[N
m]
β[d
eg]
time [s]
(a) Controller performance un-
der abrupt wind speed varia-
tions.
25 50 75 100 125 1500
1
2
3Actuated pitch angle
25 50 75 100 125 15002468
Reference angular speed
25 50 75 100 125 1500
1.5
3.55
x 104 Generator torque and bound (red)
25 50 75 100 125 150−0.05
−0.0250
0.0250.05
Angular speed error
25 50 75 100 125 1500
10
20
30
Mean wind speed
V[m
/s]
ω∗
[rad/s]
ω[rad/s]
TG,TG
sat[N
m]
β[d
eg]
time [s]
(b) Controller behavior under
wind step producing generator
torque saturation.
0 1000 2000 3000 40000
5
10
15 Actuated pitch angle
0 1000 2000 3000 40000
4
8
12Reference and optimal angular speed (red)
0 1000 2000 3000 40000
1.53
5x 10
4 Generator torque and bound (red)
0 1000 2000 3000 40000
1
2
3x 10
5 Generator Powerr
0 1000 2000 3000 400005
101520
Wind speed
V[m
/s]
ω∗,ωopt[rad/s]
PG
[W]
TG,TG
sat[N
m]
β[d
eg]
time [s]
(c) MPPT algorithm combined
with discrete time controller per-
formance.
Figure 6.4: Simulation results for a wind energy conversion system benchmark.
137
Chapter 6. SATURATED SPEED CONTROL OF MEDIUM POWER WIND TURBINES
In Fig. 6.4(a) up and down high wind steps have been reproduced, in order to test the
stability domain of the proposed solution, it’s further to notice that pejorative conditions
with respect to those considered for stability analysis in 6.3.2 has been considered; a slowly
varying reference speed covering the entire nominal range has been adopted, and a wind
step from 10 to 30 m/s, causing an initial error χ(tstep) > χmax has been produced. The
reference tracking is ensured even when an abrupt wind speed increase occurs and the
power limit is reached, causing the generator torque to drop, hence it can be reasonably
assumed that the stability properties of the proposed solution go beyond the basin of at-
traction theoretically estimated in 6.3.2.
In Fig. 6.4(b) the benefits produced by taking into account system thermal dynamics,
mentioned in 6.2.2, are clearly shown. Starting from a wind speed not requiring any
torque-power limitation, a wind step, such that the torque saturation limit is exceeded,
is produced. It can be noted how the generator torque reaches higher values than the
maximum RMS value TGRMSmax for few seconds, then, when the torque dynamic bound
decreases due to receding horizon thermal constraints, the proposed controller starts pitch
angle variation with a quite smooth trajectory, according to time-varying torque bound.
Finally, Fig. 6.4(c) reports the results obtained integrating the MPPT algorithm sketched
in 6.4, with the discrete-time controller. In order to test the reliability of the MPPT
algorithm, realistic turbulent wind speed conditions have been reproduced, by adding a
stochastic component, generated according to the widely accepted Von Karman spectrum
representation ([38]), to the wind speed mean value. It can be noted how the optimal an-
gular speed ωopt =V λopt
R is tracked quite accurately by the perturb and observe algorithm
when the wind speed is lower then the rated value, then, when power saturation, caused
by the wind step at time 2000s, occurs, the angular speed is steered to the constant value
ω = 7.6 rad/s to reduce generator power losses, while the torque-pitch coordinated action
limits the captured power. During the last part of the simulation the wind speed drops
below the rated limit and the MPPT algorithm is restarted to track the new optimal power
value.
138
Part III
Adaptive Nonlinear Estimation of
Power Electronic and
Electromechanical Systems
Parameters
139
Chapter 7
Polar Coordinates Observer for
Robust line grid parameters
estimation under unbalanced
conditions
In this chapter adaptive observers are designed to cope with the problem of es-
timating amplitude, phase and frequency of the main component of three-phase
line voltage, under unbalanced conditions. Different solutions, corresponding to
particular reference frame selections are discussed, the convergence properties
are formally proved, and a careful sensitivity analysis w.r.t harmonic distortion
and the so-called negative sequence voltage components, generated by voltage un-
balancing, is carried out. In this respect is is showed how a nonlinear adaptive
solution obtained exploiting a synchronous coordinates set can improve robustness
to unbalancing with respect to traditional solutions
7.1 Problem statement
Accurate three-phase line voltage information is required for high performance control of
power electronic applications, in particularly the correct reconstruction of phase-angle is of
utmost importance for control purpose. An example of such applications has already been
discussed in ch. 2 for what concerns Shunt Active Filter;indeed, both the unconstrained
current controller and the proposed anti-windup scheme have been designed exploiting a
suitable coordinates transformation, which bring the system into a synchronous reference
frame aligned with the line voltage vector.
In order to perform such change of coordinates, accurate informations on the line phase-
angle and frequency need to be available (see 2.7). The same considerations apply for
other power conditioning equipment such as statcom VAR compensator or Uninterrupt-
able Power Supplies (UPS). Obviously phase-angle is not available for measurement and
140
7.1. Problem statement
need to be reconstructed by elaborating the phase voltage signals. The estimation method
should be able to accommodate frequency fluctuations, not so uncommon in industrial en-
vironments, and could be expected to be even more significant in the next generation
more complex and smart grid networks. Furthermore the estimate has to be robust to
source voltage disturbances. Beside harmonics distortion, a typical grid condition to deal
with in power applications is voltage unbalance ([121]); this situation occurs when several
single-phase loads are connected to a distribution system, the fluctuating power required
by each of these loads can produce unbalance in power system, moreover, if a voltage sag
takes place in one or two phases of a three-phase power system, it produces a temporary
unbalancement [122].
Representing the three phase system with the method of the symmetrical components, it
is possible to show that voltage unbalance generates voltage terms rotating with oppo-
site phase respect to the mains voltage, for this reason they are usually called negative
sequence or counter rotating components [123]. Even though the European regulation
limits the amount of supply voltage sags ([124]), and several countries introduced specific
power quality regulation ([125], [126]), the sensitivity to negative sequence disturbances
has to be considered in order to accomplish the estimation accuracy needed in most of the
applications.
Various solutions are commonly employed for phase-angle estimation; Phase Locked Loop
(PLL) based solutions are broadly adopted, however, from a control theory standpoint,
also several estimation algorithms have been proposed. In [127] a least squares estimation
algorithm is presented, while in [128] a sensorless estimation algorithm for a specific PWM
rectifier is proposed.
Here nonlinear adaptive observers of the line voltage main component are considered;
as mentioned, beside adaptation with respect to modifications of line frequency (slowly-
varying variations of few percents around the nominal value are admissible), the key
estimation objectives is to ensure high selectiveness with respect to line voltage harmonics
and, at the same time, robustness to negative sequence at line frequency. Two different
solutions, first proposed in [43], are presented. The first one is straightforwardly derived
by LTI model of the line voltage in the so-called “stationary reference-frame”, adding a
suitable adaptation law for the line frequency. By means of mathematical analysis and
simulations, it is shown that, adopting pseudo-linear techniques, it is hard to achieve at
the same time robustness to negative sequence and good selective behavior with respect
to. harmonics. The second solutions exploits nonlinear line voltage model expressed in a
generic “synchronous reference frame” which is enforced to be aligned to the actual main
voltage vector by means of suitable adaptation laws. It will then be showed how, in this
case, an easy tuning can be performed in order to guarantee both high selectiveness and
robustness to negative sequence.
The chapter is organized as follows. In Section 7.2 the line voltage models are recalled,
adding the negative sequence representing unbalanced conditions, and the observer objec-
tive are defined. In Section 7.3 the first adaptive solution, based on stationary reference-
141
Chapter 7. POLAR COORDINATES OBSERVER FOR ROBUST LINE GRID PARAMETERS ESTIMATION
UNDER UNBALANCED CONDITIONS
frame representation is presented along with a detailed analysis on its selectiveness and
robustness with respect to the above defined voltage disturbances. In Section 7.4 the
nonlinear adaptive solution, referred to synchronous reference frame representation of line
voltage, is presented with design and stability analysis details. Also in this case the ro-
bustness and selectiveness properties are carefully discussed via analysis and simulations.
7.2 Line voltage model and adaptive estimation problem
statement
An ideal three phase source voltage system is composed by a balanced tern xa = Vmcos(ωt),
xb = Vmcos(ωt+2π3 ), xc = Vmcos(ωt+
4π3 ), according to standard two-phase planar rep-
resentation of three-phase terns the following model for xa, xb, xc can be defined
xα = −ωxβ , xα(0) = Vm
xβ = ωxα, xβ(0) = 0
xa
xb
xc
=
1 0
−12
√32
−12 −
√32
[
xα
xβ
] (7.1)
this model is usually referred to as two-phase line voltage representation in stationary ref-
erence frame (see Fig. 7.1). Differently, introducing a further coordinates transformation
[
xα
xβ
]
= T (θ)
[
xdxq
]
, T (θ) =
[
cos(θ) sin(θ)
−sin(θ) cos(θ)
]
(7.2)
where T (θ) is a time varying rotation matrix with θ = ω and θ(0) = 0, a rotating reference
frame having the d-axis aligned with the voltage vector is obtained. Therefore the so-called
line voltage representation in synchronous reference frame (referred as d − q) is derived,
and bard, q stand respectively for direct and quadrature axis (see Fig. 7.1). It is further to
recall that such coordinates set are commonly exploited in power electronic applications,
e.g. they have already been used in ch. 2 to represent the shunt active filter state variables.
Here these models are extended to describe unbalanced line voltages; if a negative sequence
(also denoted as counter-rotating component) arises, the line voltage phases change as
follows: xa = Vmcos(ωt)+Vmcrcos(−ωt+ϕ), xb = Vmcos(ωt+2π3 )+Vmcrcos(−ωt+ 2π
3 +ϕ),
xc = Vmcos(ωt +4π3 ) + Vmcrcos(−ωt + 4π
3 + ϕ), hence the stationary and synchronous
reference frame representations can be rewritten as
xα = −ωxβ xαcr = ωxβcr
xβ = ωxα xβcr = −ωxαcr
xa
xb
xc
=
1 0
−12
√32
−12 −
√32
([
xα
xβ
]
+
[
xαcr
xβcr
]) (7.3)
142
7.3. Standard adaptive observer in a two-phase stationary reference frame
a
b
c
x
xcr
−ω
ω
dq
α
β
Figure 7.1: Geometric representation of the three-phase line voltage models.
xd = 0
xq = 0[
xdcrxqcr
]
=
[
0 2ω
−2ω 0
][
xdcrxqcr
] (7.4)
xd = 0, xd(0) = Vm
xq = 0, xq(0) = 0
xa
xb
xc
=
1 0
−12
√32
−12 −
√32
[
cos(θ) −sin(θ)sin(θ) cos(θ)
][
xdxq
]
.
(7.5)
Line voltages xa, xb, xc are usually measurable, then xα, xβ in (7.1), in ideal conditions,
or xα+xαcr, xβ+xβcr in (7.3), under unbalanced conditions, can be assumed equivalently
measurable.
Relying on these line voltage vector representations, different types of phase-angle observer
can be realized, in general the following objectives have to be accomplished
1. Capability to track the grid frequency in the range of variation specified for supply
systems;
2. Selective behavior to reject voltage harmonics components;
3. Robustness to negative sequence component to properly work under unbalanced
conditions.
7.3 Standard adaptive observer in a two-phase stationary
reference frame
A natural starting point for an adaptive estimation scheme, is to consider a LTI observer
related to stationary reference-frame model (7.1), and then add angular frequency adap-
tation (assuming small variations of actual frequency with respect to the nominal one).
143
Chapter 7. POLAR COORDINATES OBSERVER FOR ROBUST LINE GRID PARAMETERS ESTIMATION
UNDER UNBALANCED CONDITIONS
The resulting adaptive observer can be expressed as
˙xα = −ωxβ + να
˙xβ = ωxα + νβ
˙ω = νω
(7.6)
where νω is the adaptation law to be designed, while να and νβ are stabilizing terms.
Defining the estimation errors xα = xα− xα, xβ = xβ − xβ, ω = ω− ω, the error dynamics
are given by˙xα = −ωxβ − ωxβ + ωxβ − να
˙xβ = ωxα + ωxα − ωxα − νβ
˙ω = −νω
(7.7)
then the following result, establishing also a tuning rule for the adaptation laws, can be
claimed
Proposition 7.3.1 Consider system (7.7) and define the frequency adaptation law νω as
νω = −xαxβ + xβxα, να = k1x1 − k3x2, νβ = k2x2 + k3x2 (7.8)
then, for any positive k1, k2, k3, system (7.7) is globally asymptotically stable at the origin.
Proof Consider the following candidate Lyapunov function
V =1
2(x2α + x2β + ω2)
taking its time derivative along the trajectories of (7.7) yields
V = −xαωxβ − xαν1 + xβωxα − νβxβ − ωνω (7.9)
which, replacing νω and να, νβ with (7.8) becomes
V = −k1x21 − k2x22 ≤ 0 (7.10)
thus, global asymptotic stability follows from direct application of La Salle’s invariance
principle.
7.3.1 Simulation results and sensitivity considerations
Beside frequency adaptation, the objectives outlined in 7.2 require a selective behavior
with respect to harmonics and rejection of negative sequences. Suitable gains selection
could lead to such characteristics. For the former objective, recalling the LTI observer
which can be obtained from (7.6) with νω = 0, “low” values of k1 and k2 would lead to
large selectiveness. Hence simulations were carried out, considering an ideal voltage with
amplitude 220VRMS and angular frequency w = 2π50rad/s, while gains of the proposed
observer were set to k1 = k2 = 0.0315 and k3 = 0.6283. Fig. 7.2(a) shows the performance
obtained when the ideal line voltage is perturbed with a 100Hz harmonic, rotating in the
144
7.3. Standard adaptive observer in a two-phase stationary reference frame
1 1.02 1.04 1.06 1.08 1.1−3
−1.5
0
1.5
3Voltage estimation error
1 1.02 1.04 1.06 1.08 1.1200
250
300
350Frequency estimation error
xα[V
]xα[V
]
time [s]
(a) Voltage estimation error referred to
ideal line voltage xα and frequency es-
timation error under a 15V 100Hz har-
monic.
1 1.02 1.04 1.06 1.08 1.1−100
−50
0
50
100Voltage estimation error
1 1.02 1.04 1.06 1.08 1.1200
250
300
350Frequency estimation error
ω[rad
/s]
xα[V
]
time [s]
(b) Voltage estimation error referred to
ideal line voltage xα and frequency es-
timation error under a 15V negative se-
quence disturbance.
Figure 7.2: Estimation performances of the adaptive observer designed in the two-phase
stationary reference frame.
same direction of line voltage vector, with amplitude of 15V (very large w.r.t. practical
cases). Large robustness of the proposed solution can be noticed, according to what ex-
pected form gains selection.
For what concerns the sensitivity to negative sequence, a further simulation scenario has
been realized. In Fig. 7.2(b) the simulation results show the observer behavior under
a negative sequence disturbance of 15V. The system shows a high sensitivity to nega-
tive sequence disturbances, the frequency estimation is affected by a huge error and the
voltage amplitude is far from the ideal value, with an error up to 60V. Hence clear dis-
turbance amplification can be observed. The unexpected behavior to negative sequence
disturbances can be investigated performing a linearization of the error dynamics (7.7)
at the origin, and exploiting analysis tools for LTI systems. Note that this is equivalent
to remove frequency adaptation and assume a known line angular frequency, recovering a
standard LTI solution.
A transfer function matrix representation of the linear observer is helpful to carry out the
sensitivity analysis. Considering the measured voltages xα, xβ as, possibly perturbed, in-
puts, and the estimated voltages xα, xβ as outputs, the observer model can be summarized
as follows
[
xα
xβ
]
=
[
G11 G12
G21 G22
][
xα
xβ
]
(7.11)
145
Chapter 7. POLAR COORDINATES OBSERVER FOR ROBUST LINE GRID PARAMETERS ESTIMATION
UNDER UNBALANCED CONDITIONS
20 30 40 50 60 70−40
−30
−20
−10
0
10
Mag
nitu
de (d
B)
Amplitude response to positive sequence line voltage
Frequency (Hz)
(a) Bode diagrams of the linear observer response to pos-
itive sequence.
20 50 80 110 140 170 200−10
0
10
20
30
40
50
Mag
nitu
de (d
B)
Amplitude response to negative sequence components
Frequency (Hz)
(b) Bode diagrams of the linear observer response to neg-
ative sequence.
Figure 7.3: Sensitivity analysis of two-phase stationary reference frame observer.
In order to analyze the observer harmonic response with respect to positive rotating (also
referred as positive sequence) harmonics and negative sequences, the relationships between
xα(jω) and xβ(jω) in both conditions need to be considered. In particular, for positive se-
quences xβ(jω) = e−j π2 xα(jω), while for negative sequences xβ(jω) = ej
π2 xα(jω). Hence,
considering the symmetry of the transfer function matrix, the following SISO transfer
functions can be used to analyze the sensitivity, G11 + G12e−π
2 , for positive sequences;
G11 + G12eπ2 for negative sequences. The corresponding Bode diagrams are reported in
Figs. 7.3(a)-7.3(b), respectively. From the frequency domain analysis the high sensitivity
to negative sequence components, previously highlighted for the adaptive observer, is con-
firmed. Moreover, it seems that, for this kind of observer, robustness to negative sequence
and selective response cannot be ensured with the same set of parameters. Formal proof
of such result is not available, but many attempts to change the gain parameters support
such conjecture.
7.4 Nonlinear adaptive observer in a synchronous polar co-
ordinates reference frame
In order to overcome the lack of robustness to unbalanced conditions enlighten in the pre-
vious sections for simple observers designed in the stationary reference frame. The main
idea is to define an observer in a generic rotating reference frame (referred as d− q), and
to made it to asymptotically converge towards the synchronous model (7.5) in the d − q
146
7.4. Nonlinear adaptive observer in a synchronous polar coordinates reference frame
reference frame.
In this respect, a generic reference frame rotation T (θ) is defined, similarly to what re-
ported in (7.2), with an angle θ and angular frequency ω s.t.˙θ = ω. Applying the
corresponding rotation to (7.1), the following generic d − q model of the line voltage is
obtainedxd = −(ω − ω)xq
xq = (ω − ω)xd.(7.12)
Then the following adaptive observer model (including θ and ω dynamics) is proposed
˙A = νA ω = νω + ˆω
˙ω = ηω˙θ = ω
xd = A, xq = 0
(7.13)
this solution can be alternatively thought as a sort of polar coordinates observer, where it
is worth noting that a sort of PI structure has been adopted for the θ estimation in order
to recover the initial unknown phase-angle value, while ˆω can be considered as the actual
angular frequency estimation of the proposed observer.
Defining ω = ω − ˆω, xd = xd − A, xq = xq yields the estimation error dynamics
˙xd = −(ω − νω)(xq)− νA
˙xq = (ω − νω)(xd + A)
˙ω = −ηω
(7.14)
then the following results, regarding the adaptation laws design for the estimate asymptotic
convergence can be stated
Proposition 7.4.1 Consider system (7.14) and define the adaptation terms as
ηω =1
γAxq, νω =
k1
Axq, νA = k2xd (7.15)
then, for arbitrary positive values of k1, k2, γ, the origin of system (7.14) is asymptotically
stable.
Proof Consider the following candidate Lyapunov-like function
V =1
2(x2d + x2q + γω2), γ > 0
its time derivative along the trajectories of (7.14) is
V = (ω − νω)Axq − ωγηω − νAxd
replacing the control terms νω, ηω, νA defined in (7.15) the time derivative of V becomes
V = −k1x2q − k2x2d (7.16)
which is negative semi-definite for arbitrary positive values of k1, k2. Thus xd, xq asymp-
totically tend to zero. Then, from direct application of Barbalat’s lemma ([15] ch. 8), it
can be concluded that limt→∞ V = 0 therefore limt→∞ ω = 0, and the equilibrium point
at the origin of the error system (7.14) is asymptotically stable.
147
Chapter 7. POLAR COORDINATES OBSERVER FOR ROBUST LINE GRID PARAMETERS ESTIMATION
UNDER UNBALANCED CONDITIONS
7.4.1 Gains selection, sensitivity analysis and simulations
In order to select a set of gains k1, k2, γ which ensures selectivity and rejection to counter
rotating components (i.e. negative sequences), the system (7.14) is linearized around the
origin, assuming the estimated amplitude A to have the correct value. Two disturbance
components dd, dq are added to the system and considered as inputs, in this way the
response to negative sequence can be characterized. Bearing in mind this consideration,
and using (7.15), the linearized model reads as
d
dt
xd
xq
ω
=
−k2 0 0
0 −k1 A
0 − Aγ 0
xd
xq
ω
+
−k2 0
0 −k10 − A
γ
[
dd
dq
]
. (7.17)
Differently from the solution presented in 7.3, thanks to the diagonal block structure of the
state matrix in (7.17), the parameter k2 can be separately designed from the parameters
k1, γ. This property allows to obtain strong selectivity and negative sequence rejection at
the same time. A low value for the parameter k2, which characterize the error dynamic
xd , is adopted in order to have a selective response. Errors on xq and frequency are
characterized by a second order dynamic which can be varied by means of k1 and γ. This
gains value can be chosen to set the resonance frequency much lower than the mains volt-
age frequency, and to introduce an attenuation of about 20dB for the negative sequence
disturbances.
Similarly to what in 7.3.1, a frequency response analysis can be carried out considering
the relation between dd and dq in the case of interest. In particular dq(jω) = e−j π2 dd(jω)
for positive rotating harmonics and dq(jω) = ejπ2 dd(jω) for negative sequences. The fol-
lowing gains have been selected k1 = 1, k2 = 1 × 10−4, γ = 10 for the observer, while
the same line parameters considered in 7.3.1 have been adopted. Owing to the structure
of the linearized model and the selected gains, Bode diagrams with respect to positive
rotating harmonics and negative sequences are very similar. In Fig. 7.4(a), 7.4(b) fre-
quency response to negative sequences of xq, and ω are reported, respectively. Resonance
is present at very low frequencies, while in the range of interest, around the line angular
speed 50Hz, and its multiples, very large attenuation is provided. The frequency response
of xd is not reported since, being completely decoupled from the other variables, can be
arbitrary shaped and it is obviously less critical.
Simulations of the nonlinear adaptive observer with the proposed gain selection has been
performed, the same conditions of 100Hz voltage harmonic and negative sequence pertur-
bation, described in 7.3.1 have been considered. The estimate ˆω has been initialized to
2π40 rad/s., while the voltage amplitude estimate A has been initialized to the 80% of
its actual value. Figs. 7.5(a)-7.5(b) show the observer behavior under a 100Hz positive
rotating harmonic and a negative sequence component, respectively. As expected from
the analysis of the linearized system, a strong rejection of the voltage disturbances can be
noticed in both conditions. Therefore, this observer scheme results much more suitable
to work under unbalanced three phase systems than the one derived from the two phase
148
7.4. Nonlinear adaptive observer in a synchronous polar coordinates reference frame
100
101
102
103
104
105
−100
−80
−60
−40
−20
0
20
40
Mag
nitu
de (d
B)
Amplitude response of voltage error on q axis to voltage disturbances
Frequency (Hz)
(a) Bode diagrams of transfer function between xq and
voltage disturbances.
100
101
102
−40
−30
−20
−10
0
10
20
30
Mag
nitu
de (d
B)
Amplitude response of frequency error to voltage disturbances
Frequency (Hz)
(b) Bode diagrams of transfer function between ω and
voltage disturbances.
Figure 7.4: Sensitivity analysis of the polar coordinates observer.
stationary representation of line voltage.
The approach can be further extended to accomplish a perfect rejection of oltage un-
balancing effects, by augmenting the synchronous observer dynamcis with the negative
sequence model (7.3). Redefining the adaptation laws, that would be augmented with two
additional terms related to the negative sequence vector estimation, it is possible to sepa-
rate the main component of the perturbed measured voltage from the the counterrotating
term given by the unbalancing (see [43] for further details).
149
Chapter 7. POLAR COORDINATES OBSERVER FOR ROBUST LINE GRID PARAMETERS ESTIMATION
UNDER UNBALANCED CONDITIONS
1 1.02 1.04 1.06 1.08 1.1−0.01−0.005
00.0050.01
Estimation error on d component
1 1.02 1.04 1.06 1.08 1.1−2−1012
Estimation error on q component
1 1.02 1.04 1.06 1.08 1.1−2−1012
Frequency estimation error
1 1.02 1.04 1.06 1.08 1.1−5
−2.50
2.55
Estimated phase−angle
xd[V
]xq[V
]ω
[rad/s]
θ[rad]
time [s]
(a) Nonlinear estimator performance
under a 100 Hz 15 V voltage harmonic
disturbances (voltage estimation errors
are calculated w.r.t. nominal voltages,
not the perturbed ones).
1 1.02 1.04 1.06 1.08 1.1−0.01−0.005
00.0050.01
Estimation error on d component
1 1.02 1.04 1.06 1.08 1.1−2−1012
Estimation error on q component
1 1.02 1.04 1.06 1.08 1.1−2−1012
Frequency estimation error
1 1.02 1.04 1.06 1.08 1.1−5
−2.50
2.55
Estimated phase−angle
time [s]
xd[V
]xq[V
]ω
[rad/s]
θ[rad]
(b) Nonlinear estimator performance
under a 15V negative sequence distur-
bance (voltage estimation errors are cal-
culated w.r.t. nominal voltages, not the
perturbed ones).
Figure 7.5: Polar coordinates observer estimation performances.
150
Chapter 8
Polar Coordinate Observer for
Position and Speed estimation in
Permanent Magnet Synchronous
Machines
In this chapter the polar coordinates observer framework is extended and specified
for rotor speed and position reconstruction in Permanent Magnet Synchronous
Machines. The reference frame adopted in the observer is pushed toward the
synchronous one by forcing it to be intrinsically aligned with the estimated back-
emf vector. This approach does not require model-based stator flux dynamicsfor
estimation, leading to improved robustness properties w.r.t measurement uncer-
tainties. Stability analysis is carried out by using singular perturbation approach.
Effective tuning guidelines are drawn exploiting insightful linearization of the
nonlinear adaptive observer.
8.1 Introduction
Permanent Magnet Synchronous Machines (PMSM) are a class of electrical machines com-
monly used for a wide range of applications. Vector control methods are usually adopted
to ensure an efficient regulation; they require the knowledge of the rotor position, and,
when a speed loop is implemented, the rotor speed feedback is also needed.
The desire of reducing the cost and the number of components, improving, at the same
time, the system reliability, has stimulated the research towards sensorless control al-
gorithms. A natural approach is to augment the system with an observer and feed the
controller with the estimated variables.
An intense research activity has been carried out to cope with this issue, some monographs
have been dedicated to it ([129], [130]) where nonlinear and adaptive control solutions have
been applied to electrical drives regulation. Despite the topic is quite mature, there is still
151
Chapter 8. POLAR COORDINATE OBSERVER FOR POSITION AND SPEED ESTIMATION IN
PERMANENT MAGNET SYNCHRONOUS MACHINES
research activity aimed to improve estimation performance under some well known critical
conditions. Two main methodologies can be outlined in the specific literature; the first,
usually referred as signal-based, includes all the approaches based on high frequency volt-
age signals injection, used to get complete position information by exploiting the magnetic
saliency (see [131], for instance). The second category, usually referred as model-based,
covers methods where nominal models of PMSM are exploited in different ways to re-
construct the rotor magnet position and speed through the back-emf induced on stator
windings.
Solution belonging to this family are commonly preferred, especially in medium or high
speed range of operation. In fact, it is well known ([132], [133]) that at low speed values,
the performance of model-based methods abruptly degrade due to a lack of observabil-
ity of the system. Another common problem of these approaches is the sensitivity to
parameters uncertainty, again particularly relevant at low speed. These two drawbacks
become even more significant when a linear approximation of the machine model is taken
to design the estimation system, hence the research effort has been devoted to develop
nonlinear observers for this kind of application. Beside the classical solutions based on
open-loop integrations of some system dynamics (typically the stator flux dynamics) or
on extended Kalman filters (see [134] among the others), some other interesting solutions,
aimed to cope with the above stated issues, have been proposed. Significant approaches,
concerning analysis and improvement of robustness with respect to parameters uncertain-
ties, have been presented along with some discussions on the stability properties of the
adopted nonlinear schemes (see [135], [136], [137]). Recently, a pair of solutions, presented
in [132] and [138] respectively, have cast the estimation problem into modern nonlinear
observer design techniques ([139], [140]), in order to provide a rigorous formal stability
analysis.
Here a novel and simple position and speed observer for PMSM, first formulated in [44]
is described. Taking the cue from the approach presented in 7.4, the idea is to build an
observer in a generic reference frame, imposing a representation for the beck-emf vector
equivalent to the one it would have in the so-called synchronous coordinates, which for
PMSM is commonly selecte to be aligned with beck-emf vector itself. At the same time the
beck-emf amplitude has to be reconstructed from the measured stator currents, along with
the rotor speed and position. The main advantage of this solution is that no pure inte-
gration of the stator dynamics is required, since the stator current dynamics are exploited
as indirect measurement of the back-emf vector. This leads to an intrinsic robustness to
many kinds of voltage and current measurement uncertainties. Time scale separation be-
tween the stator current dynamics and the remainder of the observer dynamics is induced
to provide practical semiglobal asymptotic stability.
The chapter is organized as follows. In Section 8.2 the standard model of PMSM in the
so-called synchronous reference frame is recalled and the general objectives for position
and speed sensorless reconstruction are reported. In Section 8.3 the proposed observer
is presented and stability analysis is carried out using singular perturbation approach.
152
8.2. PMSM Model and estimation problem definition
Then, similarly to what in 7.4.1, some tuning guidelines are derived by insightful analysis
of the estimation error linearized dynamics. Section 8.5 ends the chapter with signifi-
cant simulation tests that testify the properties of the proposed method under different
scenarios.
8.2 PMSM Model and estimation problem definition
According to standard planar representation of three-phase electric motors [129], the
PMSM electro-magnetic model can be represented in a generic 2-phase u-v reference frame
rotated by an angle ǫ0 with respect to a static reference frame aligned to the stator wind-
ings
iu = −RLiu + ω0iv +
ωφvL
+uu + du
L
iv = −RLiv − ω0iu − ωφu
L+uv + dvL
φu = −(pω − ω0)φv
φv = (pω − ω0)φu
(8.1)
where ω0 = ǫ0 is the angular speed of the arbitrary selected reference frame u-v; p are
the pole pairs; R,L are stator winding resistance and inductance; ω is the actual rotor
mechanical speed; iu, iv are the stator currents; φu, φv are the components given by the
projection in the considered reference frame of the rotor magnet flux vector, whose am-
plitude will be indicated as Φ. In this framework, θ, such that θ = ω and θe, such that
θe = pω, can be used respectively to represent the mechanical and the so-called electrical
angle of the rotor magnet flux vector with respect to the static stator-aligned reference
frame. Finally, uu + du and uv + dv give the voltages applied to the stator windings.
Usually, stator voltages are actuated by means of switching power converters (such as in-
verters) and direct measurements are not available or not highly accurate. For this reason
the voltages have been split into the sum of the ideal expected values uu, uv and two
terms du, dv which account for measurement errors and/or inverter non-idealities (such as
Dead-Time effect, IGBT/MOS voltage drop).
With no loss of generality, one polar pair is assumed, therefore ω will be the so-called
electrical rotor speed and the mechanical angle, θ, and the electric one, θe, will be the
same.
As it is well known, [129], defining a reference frame d-q with the d-axis aligned with the
rotor flux vector, the model (8.1) reads as follows
id = −RLid + ωiq +
ud + ddL
iq = −RLiq − ωid −
ωΦ
L+uq + dqL
φd = 0
φq = 0
(8.2)
153
Chapter 8. POLAR COORDINATE OBSERVER FOR POSITION AND SPEED ESTIMATION IN
PERMANENT MAGNET SYNCHRONOUS MACHINES
For this kind of coordinate frame, the speed ω0 and angle ǫ0 become exactly the electrical
rotor speed ω and the rotor flux vector angle θ. Therefore, similarly to what in 7.2 for
three-phase line voltage representation, this reference frame is usually referred to as syn-
chronous reference frame.
In sensorless control of PMSM a fundamental issue is to estimate the rotor flux vector angle
θ and speed ω, since no direct measurements are available. This is crucial to build stan-
dard and also some kind of advanced speed-torque controllers based on field-orientation
principles, [129]. Usually, in model-based approaches, speed and position estimation task
is performed by defining a suitable observer exploiting the electromagnetic model of the
PMSM, while no relevant information is assumed available on the mechanical model of
what connected to the machine rotor. On the other hand, the speed dynamics is as-
sumed much slower than the electromagnetic one, therefore the speed is assumed constant
(or slowly varying) in stating the above-mentioned estimation problem. Beside the basic
problem, also the estimation of the amplitude of the rotor magnetic flux vector is often
considered to enable very accurate torque control accounting for flux amplitude variations
along time or due to different working temperatures [129]).
Bearing in mind these considerations, the following general objectives can be defined for
an observer based on the electromagnetic model of PMSM.
1. Guaranteeing estimation of rotor magnet vector position, θ, and speed ω along with
its amplitude Φ, under constant rotation conditions, assuming stator currents and
expected stator voltages available from measures and actuations, respectively, and
considering null voltage uncertainties (these conditions will be referred as nominal
conditions);
2. Achieving as large as possible bandwidth in the estimation of the speed ω in order
to compensate for the lack of knowledge of the mechanical model and cope with
variable speed conditions;
3. Obtaining large voltage disturbances rejection, i.e. attenuation of the dd, dq distur-
bances.
8.3 Nonlinear adaptive observer based on a synchronous
reference frame
The basic idea of the proposed approach, is similar to what already showed in (7.4). Here
the observer is built by imposing in a generic reference frame the model (8.2) which is
valid only in the synchronous one, then feedback estimation laws are designed in order to
push the angle and the speed of the adopted reference toward θ and ω of the synchronous
frame. Therefore, the proposed observer reference frame can be seen as an estimation of
the synchronous one and, in the following, it will be denoted as d - q with angle θ and
speed ω with respect to static stator reference frame.
154
8.3. Nonlinear adaptive observer based on a synchronous reference frame
An additional important step in the above defined procedure is a coordinate change high-
lighting the back-emf as state variable. For this purpose define χd = ωφd and χq = ωφq,
then the synchronous model (8.2) can be revised leading to the following observer in the
d - q reference frame˙id = −R
Lid + ωiq +
udL
+ ηd
˙iq = −RLiq − ωid −
A
L+uqL
+ ηq
˙A = νa
˙ω = ηω
ω = ˆω + νω
˙θ = ω
(8.3)
where id, iq and ud, uq are the stator currents and expected voltages, available from
measurements, and the actuator commands and reported in d-q frame; A is the estimation
of the back-emf ωΦ in (8.2); while the meaning of id, iq, θ and ω is straightforward.
Differently, ηd, ηq, νa, ηω and νω are feedback terms defined as follows, for observer
convergence
ηd = kpid, ηq = kpiq, νa = −Lk1kpiq,
ηω = γA
Lkpid, νω = k2
A
Lkpid
(8.4)
where id = id − id, iq = iq − iq. As in (7.13), a sort of PI structure has been adopted
for the estimation of θ, but just ˆω will be considered as output speed estimation of the
proposed observer. Finally, the PMSM model can be expressed in the observer reference
frame as follows
id = −RLid + ωiq +
χq
L+ud + ddL
iq = −RLiq − ωid −
χd
L+uq + dqL
χd = −(ω − ω)χq +ω
ωχd
χq = (ω − ω)χd +ω
ωχq
(8.5)
where χd χq underscore the back-emf projections in the considered frame, according to
the previous definitions.
Note that in model (8.5) also dd, dq and ω are reported to highlight the effect of voltage
uncertainties and non-constant speed conditions.
8.3.1 Stability analysis
The convergence analysis of the proposed estimation scheme will be carried out assuming
nominal conditions defined in Objective 1 at the end of 8.2, hence the disturbances on
the actuated voltages and the perturbation introduced by a non constant rotor speed,
appearing in (8.5), will be neglected. However, these additional input signals will be
155
Chapter 8. POLAR COORDINATE OBSERVER FOR POSITION AND SPEED ESTIMATION IN
PERMANENT MAGNET SYNCHRONOUS MACHINES
considered in 8.4 for observer gains tuning, according to objectives 2-3 stated at the end
of 8.2.
A model to suitably represent the behavior of the observation error can be defined by
considering the current errors id and iq, previously introduced, and adding the following
errors variables related to the estimation of the back-emf components and speed
χd = χd − A , χq = χq , ω = ω − ˆω (8.6)
By subtracting (8.3) from (8.5), disregarding dd, dq, ω, the dynamics of the above defined
estimation errors is the following
˙id = −ηd +χq
L
˙iq = −ηq −χd
L˙χd = −(ω − νω)χq − νa
˙χq = (ω − νω)(χd + A)
˙ω = −ηω
(8.7)
Exploiting the adaptation laws defined in (8.4) and defining the following change of co-
ordinates χd1= χd/Lkp, χq1 = χq/Lkp, νa1 = νa/Lkp, A1 = A/Lkp and ǫ = 1
kp, system
(8.7) reads as
ǫ˙id = −id + χq1
ǫ˙iq = −iq − χd1
˙χd1= −(ω − k2A1id)χq1 + k1iq
˙χq1 = (ω − k2A1id)(χd1+ A1)
˙ω = −γA1id
(8.8)
This can be easily seen as a standard singular perturbation model ([15] ch. 11), where
the time scale separation between the current error dynamics and the back-emf and speed
error dynamics is parametrized by the gain kp. Therefore, assuming a sufficiently high
value of kp has been chosen (more details will be given in the following), the problem
of the estimates convergence can be approached by considering the overall system as the
interconnection of a fast subsystem, represented by the current error variables (id ,iq), and
a slow subsystem given by the other dynamics (χd1, χq1 , ω).
According to [15] and [76], we start by studying the so-called boundary layer system,
related to the fast dynamics. First, define the quasi steady-state value for the current
errors, obtained as the solution of the fast subsystem when ǫ = 0, it results id = χq1(t),
iq = −χd1(t). Then, defining yd = id − χq1 , yq = iq + χd1
, t = ǫτ , after some computation,
consisting in freezing the slow varying variables by setting ǫ = 0, the following boundary
layer system is obtaineddyddτ
= −yd,dyqdτ
= −yq (8.9)
It is trivial to verify that the origin of (8.9) is globally exponentially stable, uniformly
in both the slow variables and the time, since it is a LTI system with an Hurwitz state
156
8.4. Tuning rules for the proposed solution
matrix.
Now, the focus is put on the reduced dynamics obtained by substituting the fast variables
id, iq with their quasi steady-state, id = χq1(t), iq = −χd1(t), in the slow dynamics given
by the last three equations in (8.8). Note that the quasi steady-state definition enlightens
how the current errors can be used as indirect measure of the back-emf estimation errors,
thanks to time scale separation imposed by kp. After some computation the following
reduced system results
˙χd1= −(ω − k2A1χq1)χq1 − k1χd1
˙χq1 = (ω − k2A1χq1)(χd1+ A1)
˙ω = −γA1χq1
(8.10)
To investigate the stability of (8.10) consider the Lyapunov candidate function V = 12(χ
2d1+
χ2q1+ ω2
γ ), taking its derivative along the system trajectories yields
V = −k1χ2d1
− k2A21χ
2q1 (8.11)
which is negative semi-definite for any positive values of k1, k2. Therefore, from direct
application of Barbalat’s lemma, it can be stated that limt→∞ V = 0, limt→∞ ˙χd1= 0 and
limt→∞ ˙χq1 = 0. Therefore, the origin of the reduced dynamics is globally asymptotically
stable.
From the previous considerations and using the singular perturbation results as formulated
in [76], the following properties for the overall estimation error dynamics (8.8) can be
drawn.
Proposition 8.3.1 For the system (8.8), replacing for simplicity current coordinates, id,
iq, with the above defined yd = id − χq1, yq = iq + χd1, there exist two class KL functions
βf and βs such that, for each δ > 0 and for every compact sets Ωf ⊂ R2 and Ωs ⊂ R
3,
there exists ǫ∗ such that ∀ǫ = k−1p ∈ (0, ǫ∗], the following inequalities hold
‖[yd(t), yq(t)]T ‖ ≤ βf(‖[yd(0), yq(0)]T ‖, t/ǫ
)+ δ ∀[yd(0), yq(0)]T ∈ Ωf (8.12)
‖[χd1(t), χq1(t), ω(t)]
T ‖ ≤ βs
(
‖[χd1(0), χq1(0), ω(0)]
T ‖, t)
+ δ ∀[χd1(0), χq1(0), ω(0)]
T ∈ Ωs
(8.13)
Hence, semiglobal practical stability can be stated for the overall error dynamics (8.8),
provided that a sufficiently large kp has been selected.
8.4 Tuning rules for the proposed solution
The time scale separation properties derived in 8.3.1 and the general objectives defined in
8.2 can be induced applying some gain tuning guidelines for the adaptation laws of the
proposed observer. A preliminary step toward this goal is to rewrite the error dynamics
157
Chapter 8. POLAR COORDINATE OBSERVER FOR POSITION AND SPEED ESTIMATION IN
PERMANENT MAGNET SYNCHRONOUS MACHINES
(8.7) taking into account the voltage disturbances and the perturbation given by non-
constant speed as follows
˙id = kp(−id + χq1) +ddL
˙iq = −kp(iq + χd1) +
dqL
˙χd1= −(ω − νω)χq1 − νa1 +
ω
ω(χd1
+ A1)
˙χq1 = (ω − νω)(χd1+ A1) +
ω
ωχq1
˙ω = −ηω + ω
(8.14)
The origin of the system, [χd1χq1 ω] = 0, is an equilibrium point, and linearizing the
system near the origin the following LTI system is obtained
˙id = kp(−id + χq1) +ddL, ˙iq = kp(−iq − χd1
) +dqL
˙χd1= −νa1 + ωΦ1
˙χq1 = (ω − νω)Φ1ω (8.15)
˙ω = −ηω + ω
where Φ1 = Φ/Lkp. The variable ω can be seen as an input acting on ˙ω and ˙χd1and it
is useful to evaluate the sensitivity of the error variables to the variable speed, i.e. the
observer bandwidth. Other inputs in the dynamics (8.15) are the voltage disturbances ddand dq, also the sensitivity to such variables will be considered.
Applying to the error system (8.15) the results deriving from singular perturbation prop-
erties enlightened in 8.3.1, the following quasi-steady state equations can be considered
(with some abuse of notation)
−id + χq1 +ddLkp
≈ 0, −iq − χd1 +dqLkp
≈ 0 (8.16)
Hence, the following linearized reduced error dynamics can be derived
˙χd1= k1(−χd1
+ dq/Lkp) + ωΦ1
˙χq1 = ωΦ1ω − k2(ωΦ1)2 (χq1 + dq/Lkp)
˙ω = −γωΦ1
(χq1 + dd/Lkp
)+ ω
(8.17)
with the following state matrix AR and input matrix BR with respect to the input vector
[dd dq ω]T
AR =
−k1 0 0
0 −k2(Φ1ω)2 Φ1ω
0 −γA1 0
, BR = 1
Lkp
0 k1 Φ
−k2(ωΦ1)2 0 0
−γωΦ1 0 Lkp
(8.18)
State matrix AR in (8.18) has the following eigenvalues
λ1 = −k1, λ2,3 =k2(ωΦ1)
2
2
[
−1±√
1− 4γ
k22(ωΦ1)2
]
. (8.19)
158
8.5. Simulation results
It is possible to find the value of k2, γ to impose damping (δ) and angular natural frequency
(ωn) for the eigenvalues λ2,3 using the following equations
k2 =2ωnδ
(ωΦ1)2, γ =
1
(ωΦ1)2
[
ω2n(1− δ2) +
k22(ωΦ1)4
4
]
(8.20)
With these formulas at hand, and bearing in mind the introduction of this section the
tuning parameter must be chosen to cope with:
1. Frequency separation, between fast dynamics of id1 , iq1 and slow dynamics of χd1,
χq1 , ω;
2. High Bandwidth Observer, for good estimation during speed variations, i.e. low
sensitivity to ω;
3. Disturbance rejection for high robustness to common disturbances due to Inverter
non-idealities, i.e. low sensitivity to dd, dq.
Obviously, frequency separation can be obtained choosing large kp. The upper bound for
this parameter is usually related to the common discrete time realization of the observer.
In fact, kp represents the bandwidth for the current id, iq reconstruction.
High observer bandwidth can be obtained acting on k1, k2 and γ but, actually, this is in
contrast with disturbance rejection requirement.
First of all, a good practice is to chose k1, k2 and γ such that they identify three distinct
eigenvalues for the matrix AR, to avoid ill conditioned problems. k1 is related only on
the bandwidth of the χd1dynamic. Its value must be chosen to be lower than the faster
dynamic imposed by kp (e.g. k1 = kp/50), recalling that a low value for this parameter
produces a low sensitivity to ω.
k2 and γ can be chosen to impose damping (δ) and angular natural frequency (ωn) of λ2,3
eigenvalues of the reduced order system, as reported in (8.20). For the value of ωn, the
same considerations as for k1 hold, i.e. ωn must be lower than the fast dynamic imposed
by kp (e.g. ωn = kp/80), but not too low to not compromise the observer bandwidth. The
damping of the eigenvalues λ2,3 (δ < 1) can be chosen to lightly augment the frequency
of the eigenvalue related to it, but its major effect is to introduce a resonant frequency
behavior, giving low disturbance rejection for a particular disturbance band frequency.
For what concerns the disturbance rejection, a preliminary task is the identification of the
disturbance band frequency. Inverter non-idealities introduce voltage disturbances with
frequencies n-times the actual electrical frequency, and from practical experiments for the
main disturbance component n = 6. The worst case is at low electrical frequency, for
which also disturbances are at low frequency.
8.5 Simulation results
In order to prove the effectiveness of the proposed approach for permanent magnet electri-
cal machines sensorless control, simulation test have been carried out plugging the observer
159
Chapter 8. POLAR COORDINATE OBSERVER FOR POSITION AND SPEED ESTIMATION IN
PERMANENT MAGNET SYNCHRONOUS MACHINES
PMSM parameters
Motor inertia J [Kgm2] 0.04
Nominal angular speed ωnom [rad/s] 300
Rotor flux Φ [Wb] 1
Nominal torque Tnom [Nm] 5
Stator resistance R [Ω] 2.5
Stator inductance L [H] 0.1
Number of pole pairs p 1
Table 8.1
in a typical field-oriented control scenario, namely the estimated speed (ˆω) and angular
position (θ) are used to feed the following standard speed controller designed in the d− q
reference frameT ∗el = kpωω + η, η = kIωω
ud = kpdid + ξd − Liq + A, ξd = kIdid
uq = kpq iq + ξq + Lid, ξq = kIq iq
id = id − i∗d, i∗
d= 0
iq = iq − i∗q , i∗q =T ∗ ˆω
A.
(8.21)
A benchmark permanent magnet electrical machine, defined by the parameters reported
in Tab.8.1, has been considered. The observer gains, tuned according to linear analysis
discussed in 8.4, are: kp = 3030, k1 = 60.6, k2 = 4503, γ = 927050. The initial estimate
value is set to zero for all the estimated variables, in order to confirm, by simulation, the
convergence property of the observer. Fig.8.1 shows the obtained results, a variable speed
reference trajectory ω∗ (see Fig. 8.1(a)) and load torque (TL in Fig. 8.1(b)) steps have
been reproduced in order to test also the observer dynamic behavior. It can be noted
how the angular speed and position estimates are quickly recovered during reference and
load torque changes, even when slow speed region is crossed, consequently also the speed
controller is able to ensure a good tracking response (see Fig. 8.1(c)). Therefore the
solution can be suitably extended and specifically tuned for realistic electrical machines
applied in timely topics such as wind energy conversion systems presented in ch. 6,
simulations tests regarding this possibility have been already carried out and reported
in [44].
160
8.5. Simulation results
0 5 10 15 20−400−200
0200400
time [s]
ω∗[rad/s]
0 5 10 15 20−400−200
0200400
time [s]
ω,ˆ ω
[rad/s]
(a) Reference, estimated and actual angular speed.
0 5 10 15 20−5
−2.50
2.55
time [s]
TL
[Nm]
0 5 10 15 20−5
−2.50
2.55
time [s]
Tel[N
m]
(b) Load and motor torque.
0 5 10 15 20−10−50510
time [s]
ω[rad/s]
0 5 10 15 20−0.5−0.25
00.250.5
time [s]
θ[rad]
(c) Speed tracking and position estimation errors.
Figure 8.1: Simulation results of sensorless speed control using the proposed observer.
161
Appendix A
Mathematical Tools
This appendix provides an overview of the mathematical tools commonly exploited to
formulate and solve the LMI-constrained optimization problems, that as reported in 1, 4,
arises in modern anti-windup solutions and explicit saturated control design techniques.
The three kind of convex and quasi-convex optimization problems that most commonly
appear in the formulations of the above mentioned problems are presented, then a simple
algorithm for their solution is introduced, and more advanced solution techniques, nowa-
days adopted to efficiently solve the considered class of problems, and generally suitable
to deal with generic nonlinear convex problems, are sketched for sake of completeness.
Finally the two mathematical tools; Schur complement and S-Procedure, deeply exploited
throughout 1, 4 and 5 to cast nonlinear inequality constraints into LMIs, will be formally
presented.
First we give some preliminary definitions; a generic LMI can be expressed as
F (x) := F0 +m∑
i=0
xiFi > 0 (A.1)
where x ∈ Rm are the decision variables, Fi are symmetric given matrix, and the in-
equality symbol means that F (x) is positive definite. Equation A.1 is a sort of explicit
representation of the LMI, while often problems are formulated letting the matrices to be
the variables, e.g. the Lyapunov equation ATP +PA < 0, with A given and P = P T as a
variable. This equation can be readily put in the form of A.1, defining a basis P1, . . . , Pm
for the m × m symmetric matrix ad taking F0 = 0, Fi = −ATPi − PiA. However, for
both notation and computational reasons, it’s often convenient to keep the LMI in its
condensed form.
In equation A.1 a strict inequality has been considered, however, in many cases, we need
to deal with non strict conditions, i.e F (x) ≥ 0. If the strict LMI F (x) is feasible, than we
say F (x) ≥ 0 is strictly feasible (constraint qualification condition, [75]), and the feasible
set of the nonstrict LMI will be the closure of the the feasible set of the strict LMI. Thus
we can simply solve the optimizations problems replacing the non-strict constraint with
its strict version. When F (x) ≥ 0 is feasible but not strictly feasible, then the previous
procedure cannot be followed, by the way, we can note that if the strict inequality is un-
162
A.1. LMI feasibility problem, Eigenvalue Problem and Generalized Eigenvalue Problem
feasible and its strict version is feasible, two possible pathological situations are present.
The first is when the non-strict inequality implicitly defines an equality constraint, the
second if the matrix F (x) has a constant nullspace (i.e F (x) is always singular). Bearing
in mind this considerations, we can state that any feasible non strict LMI can be reduced
to a strictly feasible inequality condition, by eliminating the implicit equality constraints
and removing any constant nullspace.
Formally we can say that, for any F (x) ≥ 0, there exist a matrix A, a vector b, and a set
of matrices F (z) (defined according to A.1) such that
F (x) ≥ 0 ⇔ Az + bx, F (z) ≥ 0 (A.2)
where F (z) ≥ 0 is either strictly feasible or unfeasible. The matrix A and the vector
b define the implicit equality constraints, while the matrix F (z) is the original set of
inequalities with the constant nullspace removed.
A.1 LMI feasibility problem, Eigenvalue Problem and Gen-
eralized Eigenvalue Problem
Here some standard convex and quasi-convex problems arising in control theory are pre-
sented. The most basic problem we can think about is to determine if a set of linear
matrix inequalities F (x) > 0 is feasible, i.e there exist a xfeas such that F (xfeas) > 0.
This problem is usually refereed as LMI problem, a classical example is the simultane-
ous stability problem arising for polytopic LDIs. Another common convex problem is the
so-called eigenvalue problem (EVP), where the objective is to minimize the maximum
eigenvalue of a matrix, which depends affinely on a variable, subject to an LMI constraint.
Formally a generic EVP is formulated as
minx
λ
s.t. λI −A(x) > 0, B(x) > 0(A.3)
A, B are symmetric matrices depending on the decision variable x. Equivalently an EVP
can arise in the form of minimizing a linear function subject to an LMI
min cTx
s.t. F (x) > 0(A.4)
which reduces to an LP problem if F (x) is composed with all diagonal matrices. Another
equivalent form for the generic EVP, commonly appearing in control problems is
min λ
s.t. A(λ, x) > 0(A.5)
where A is affine in (x, λ).
A third standard problem arising in control theory applications is the so-called general-
ized eigenvalue problem (GEVP) which consist in minimizing the maximum generalized
163
Chapter A. MATHEMATICAL TOOLS
eigenvalue of a pair of matrices that depends affinely on a variable, subject to an LMI
constraint. A GEVP can be expressed as
min λ
s.t. λB(x)−A(x) > 0, B(x) > 0, C(x) > 0(A.6)
where A, B, C are symmetric matrices depending affinely on x. Equivalently we can
express this as
min λmax(A(x), B(x))
s.t. B(x) > 0, C(x) > 0(A.7)
where λmax(A(x), B(x)) denotes the largest eigenvalue of the matrix B−1/2AB−1/2. Hence
we can see that this is a quasi-convex problem, since the constraint is convex (LMI) while
the objective function is quasi-convex, however, as it will be showed in the next section,
there exist reliable algorithms to solve this particular kind of nonlinear problem.
As for the EVP problem, the GEVP problem can appear in a third equivalent form
min λ
s.t. A(x, λ) > 0(A.8)
where A is affine in x for fixed λ and viceversa, furthermore it satisfies the monotonicity
condition w.r.t λ.
A.2 The ellipsoid method and interior point methods
The problems defined in the previous section can be efficiently (polynomial time) solved
with a simple algorithm based on the approximation of the region containing the optimal
point, by means of smaller and smaller ellipsoids. For this reason it is usually referred
to as the ellipsoid method. In order to present the basic insight behind the method, it
will be assumed that the problem has at least an optimal point, hence the constraints
are feasible. Roughly speaking, the idea is to start with an ellipsoid E0 containing the
optimal point. Than the ellipsoid is cut by a plane passing through its center, this means
that the optimal point will be guaranteed to lie on one of the two half-spaces defined by
the cutting plane. Hence we can compute a vector g(0) (defining the cutting plane), such
that the optimal point will lie inz | g(0)T (z − x(0)) < 0
. The next step is to compute
the smallest ellipsoid (minimum volume) E(1) containing the half-ellipsoid corresponding
to E(0) ∩ g(0)T (z − x(0)) < 0, which is guaranteed to contain the optimal point. The
procedure can then be iterated by slicing the ellipsoid E(1) with a cutting plane.
It’s further to notice that the minimum volume ellipsoid contained in an half-ellipsoid can
be expressed analytically ([75]), if we consider the generic half-ellipsoid
E =z | (z − a)TA−1(z − a) ≤ 1, gT (z − a) ≤ 0
164
A.2. The ellipsoid method and interior point methods
we can state that it’s contained in the minimum volume ellipsoid
E =
z | (z − a)T A−1(z − a)
≤ 1,
wherea = a− (Ag)/(m+ 1)
g = g/√
gTAg
A =m2
m2 − 1
(
A− 2
m+ 1AT gT gA
)
.
(A.9)
Therefore, if the algorithm is initialized with x(0) and A(0) such that the resulting ellipsoid
is ensured to contain the optimal point, we only need to define how to compute a cutting
plane at each step.
For the LMI problem, if x is unfeasible, there exist u such that
uT(
F0 +
m∑
i=1
Fi(x)
)
u ≤ 0. (A.10)
Defining the components of g as gi = −uTFiu, we can state that for any z such that
gT (z − x) ≥ 0, we have
uT(
F0 +m∑
i=1
Fi(x)
)
u ≤ uTF (x)u− gT (z − x) < 0. (A.11)
It follows that every feasible point lies in the half-spacez | gT (z − x) < 0
, and thus g
define a cutting plane at the point x for the LMI problem.
Now consider the EVPmin cTx
s.t. F (x) > 0(A.12)
if x is unfeasible we can compute g as for the LMI problem, while if x is feasible g = c
defines a cutting plane, since all the points belonging to the half-spacez | cT (z − x) > 0
have an objective function greater than x, and they cannot be optimal.
Similar reasoning can be done to define a cutting plane for the GEVP; consider the ex-
pression in eq. A.7, if x is unfeasible we know how to define the cutting plane with the
method for the LMI problem. If x is feasible, picking a vector u 6= 0 such that
(λmax(A(x), B(x))B(x)−A(x))u = 0
gi = uT (λmax(A(x), B(x))Bi(x)−Ai(x))u(A.13)
we see that g defines a cutting plane for the GEVP. Finally note that for a generic convex
problem subject to LMI constraints, a cutting plane is defined by the gradient of the objec-
tive function, this directly stems from the convexity first order condition for differentiable
functions ([91]).
The ellipsoid method is still adopted for its low simplicity, however, in the last decades
it has been outdated by more efficient interior point algorithms, among those, a pretty
popular method, exploited also to compute effective approximated solutions of non-convex
165
Chapter A. MATHEMATICAL TOOLS
problems (see [81], [82]), is the so-called method of centers. It’s based on the following
definitions; consider the LMI in equation (A.1), we define the barrier-function φ(x) as
φ(x) =
log det(F (x)−1) F (x) > 0
∞ otherwise(A.14)
assuming that the problem has a nonempty and bounded feasibility set, the above defined
function is convex, thus it has a unique minimizer x∗ = argminx φ(x), x∗ is usually referred
as the analytic center of the LMI, and it can be computed by means of standard Newton’s
method. As far as concerns the EVP problem in the form of eq. (A.12), we can considering
an equivalent feasibility problem
cTx < λ, F (x) > 0 (A.15)
which is feasible for each λ > cTxopt. Therefore we can define the analytic center for
problem (A.15) as
x∗(λ) = argminx
(
log det(F (x)−1) + log1
λ− cTx
)
(A.16)
the curve x∗(λ), for λ > cTxopt is called the path of centers, and it can be shown that the
limit of the minimizing sequence defined by (A.16), for λ = λopt is the optimal point xopt.
The method of centers, for the EVP, can be specified:
• Initialize the algorithm with x(0), λ(0) s.t. (A.15) is feasible.
• Compute λ(k+1) = (1− θ)cTx+ θλ(k)
• Compute x(k+1) = x∗(λ(k+1))
where the parameter θ lies in the range [0, 1], and it is used to ensure that the current
iterate x(k) satisfies the inequality cTx < λ(k+1).
This method reduces the EVP problem to solve a sequence of unconstrained convex prob-
lem, which is usually done by Newton’s method, allowing to exploit the particular structure
of the problem, and reducing the computational effort. However in its standard formula-
tion, the method of centers has not a polynomial convergence, but it can be made to con-
verge polynomially with some modifications, for further details about these topic, stopping
criterion and unfeasibility detection methods see [81] and reference therein. More efficient
interior point algorithms, usually referred as primal-dual algorithms can be adopted to
solve the convex problems described in this section, in [91] a rather comprehensive survey
of this kind of approach can be found.
A.3 Schur complement and S-Procedure
In this section we define two mathematical tools that are often adopted to formulate
LMI constrained optimization problem starting from standard problems arising in control
166
A.3. Schur complement and S-Procedure
system theory.
In many practical cases, convex nonlinear inequalities can be converted to LMI by using
the so-called Schur complement ([12]);
Lemma A.3.1 Given two symmetric matrices Q, R, and S having the same number of
row as Q and the same number of columns as R, then the LMI condition[
Q S
ST R
]
> 0 (A.17)
is equivalent to the nonlinear matrix condition
Q− SR−1ST > 0, R > 0 (A.18)
a typical example of the Schur comlement application regards the maximum singular value
matrix norm condition ||Z|| ≤ 1. Eventhough it’s nonlinear in the matrix variable Z, it can
be expressed as the LMI
[
I Z
ZT I
]
> 0, noting that ||Z|| ≤ 1 is equivalent to I−ZTZ > 0
and then applying the above lemma.
Another useful tools, extensively used in robust control literature is the so called S-
procedure. In some problems, we find that some quadratic function must be negative
whenever some other quadratic functions are all negative. With the S-procedure, we can
replace this problem by one inequality to be satisfied by introducing some positive scalar
variables to be determined. In it’s most generic formulation S-procedure reads as
Lemma A.3.2 Consider a family of quadratic functions Fi(ξ) in the form Fi(ξ) = ξTPiξ+
2uTi ξvi, i = 0, . . . p, where Pi = P Ti , then if there exist numbers τ1, . . . τp ≥ 0 such that
∀ξ, F0(ξ)−p∑
i=1
τiFi(ξ) ≥ 0 (A.19)
then the following holds.
F0(ξ) ≥ 0 ∀ξ s.t. Fi(ξ) ≥ 0, i = 1, . . . p. (A.20)
The nontrivial converse holds if p = 1 and F1(ξ∗) > 0 for some ξ∗.
A variation of S-procedure involving strict inequalities and quadratic forms can be estab-
lished as follows
Lemma A.3.3 Let Pi, i = 0, . . . , p be symmetric matrices, and consider the following
condition
ξTP0ξ > 0, ∀ξ 6= 0 s.t ξTPiξ ≥ 0, i = 1, . . . , p (A.21)
it’s easy to see that if there exist τ1, . . . τp ≥ 0 such that
P0 −p∑
i=1
τiPi > 0 (A.22)
then (A.21) holds. The converse is true when p = 1 if for some ξ∗, ξ∗TP1ξ∗ > 0.
167
Chapter A. MATHEMATICAL TOOLS
A.4 Finsler’s and Elimination Lemma
A property closely related to the previously presented S-procedure is the so called Finsler’s
lemma whose statement is reported in the following
Lemma A.4.1 Given two real symmetric matrices P , A if the quadratic inequality
xTPx > 0 (A.23)
holds for any x 6= 0 such that xTAx = 0, then there exist a scalar λ such that
Q− λA > 0 (A.24)
Finsler’s lemma is exploited to derive onother useful lemmma, the so called elimination
lemma [75], used to reformulate matrix inequalities eliminating some of the original vari-
ables
Lemma A.4.2 Consider the matrix inequality
T (x) +W (x)FV T (x) + V (x)F TW T (x) > 0 (A.25)
with T ∈ Rn×n, and W , V , F of suitable dimension. Assume that T , W , V are inde-
pendent from F and denote with W⊥(x), V ⊥(x) the orthogonal complements of W (x),
V (x) respectively. Then (A.25) holds for some F and x = x0 if and only if the following
inequalities are satisfied for x = x0
W⊥(x)T (x)W⊥(x) > 0
V ⊥(x)T (x)V ⊥(x) > 0.(A.26)
Furthermore, by applying lemma A.4.1 we can state that there exists λ ∈ R such that
T (x)− λV (x)V (x)T > 0
T (x)− λW (x)W (x)T > 0(A.27)
A.5 Sector characterization for saturation and deadzone non-
linearities
The most common way to deal with saturation and deadzone nonlinearities for anti-windup
purposes or to derive sufficient stabilizability and stabilization conditions for saturated
control systems is by means of the so called sector characterization stemmed from absolute
stability arguments. In general the following definition [15] can be given
Definition A memoryless nonlinearity φ(u) is said to belong to the sector [K1 K2] where
K1 = diag α1, α2, . . . , αm, K2 = diag β1, β2, . . . , βm if
(φ(q)−K1q)T (φ(q)−K2q) ≤ 0. (A.28)
168
A.5. Sector characterization for saturation and deadzone nonlinearities
then it’s easy to verify that the decentralized symmetric saturation function defined in
(1.1) and the associated deadzone function dz(u) = u− sat(u) belong respectively to the
conic sectors [0, Im], [−Im 0] (see Fig. A.1 for the geometric interpretation). Therefore
(A.28) is specialized to the following inequality for what concerns the saturation function
p = sat(u)
pTW (p− u) ≤ 0 (A.29)
with W an arbitrary diagonal positive definite matrix. While as far as the deadzone
nonlinearity q = dz(u) is concerned, the following characterization holds
(u+ q)TWq ≤ 0. (A.30)
The above conditions are pretty straightforward to prove, for the sake of completeness we
pi
ui
(a) Sector characterization of satu-
ration nonlinearity.
qi
ui
(b) Sector characterization of dead-
zone nonlinearity.
Figure A.1: Global sector conditions of saturation and deadzone nonlinearity.
motivate condition (A.29), similar reasoning can be made for (A.30). When p = sat(u) = u
then (A.29) applies with the equality, while if p 6= u, by definition (1.1), the sign of (p−u)is opposite to the sign of u which is equal to the sign of p, hence the product is always
negative. The above condition are global, that is they hold for any u ∈ Rm, however, to
obtain significant results for the regional analysis of saturated systems, it is profitable to
derive less conservative sector characterizations. Among all the fruitful ideas proposed in
the literature, here the generalized condition (1.2.1) for the deadzone nonlinearity, defined
to present the main results concerning direct linear anti-windup approach is reported along
with the proof (see [12] for other possible solutions).
For convenience we recall the condition
q(u)TS−1(q(u) + ω) ≤ 0 (A.31)
that is satisfied for for any positive diagonal matrix S ∈ Rm×m, and for any u and ω that
are elements of the set S(usat) := u ∈ Rm, ω ∈ R
m : −usat ≤ u− ω ≤ usat, and . As
showed in 1.2, the above inequality can be made either global or local depending on the
choice of the parameter ω. The proof of lemma 1.2.1 proceed as follows [141]
Proof Assume that u and ω belong to the set S(usat), then we have usat − ui − ωi ≥ 0
and −usat − ui + ωi ≤ 0, i = 1, . . . ,m. The following cases can be outlined
169
Chapter A. MATHEMATICAL TOOLS
• If ui > usat it follows q(ui) = usat − ui < 0. Hence, since by assumption Ti,i > 0, we
obtain q(ui)Ti,i(usat − ui + ωi) = q(ui)Ti,i(q(ui) + ωi) ≤ 0
• If −usat ≤ ui ≤ usat, it follows q(ui) = 0 and (A.31) holds with equality for any T
• If ui < −usat, then q(ui) = −usat − ui > 0. Hence, since by assumption Ti,i > 0, we
obtain q(ui)Ti,i(−usat − ui + ωi) = q(ui)Ti,i(q(ui) + ωi) ≤ 0
finally we can conclude that q(ui)Ti,i(q(ui) +ωi) ≤ 0 ∀i = 1, . . . ,m for any ω, u belonging
to S(usat).
170
Appendix B
Some Considerations on Practical
PI Anti-Windup Solutions
Here some practice-driven guidelines for the anti-windup of SISO proportional integral
controllers are briefly discussed, particular attention is paid to the case of non symmetric
saturation bounds. Even if the formalism reported in ch. 1 can be in principle adopted, its
non conservative extension to non symmetric bounds, i.e without shrinking the saturation
symmetric limit to the smallest value of the asymmetric ones, can be not trivial, and for
simple controller structure like a PI it is possible to derive simpler and effective specific
approaches.
The generic structure of a PI controller is recalled, underscoring the proportional part,
denoted as P , and the integral term, denote as I, for convenience
u(t) = kpx(t)︸ ︷︷ ︸
P
+
∫ t
t0
kix(τ)dτ
︸ ︷︷ ︸
I
.(B.1)
Assume the control input is constrained to range on [um, uM ] with um ≤ 0, uM ≥ 0. The
standard anti-windup approach implemented in most of the industrial applications, consist
in freezing the integral term when the control effort hit the saturation bounds, according
to the following law
IAW =
∫ tt0kix(τ)dτ if uuc ∈ [um, uM ]
satuMum
(u)− P otherwise(B.2)
where satuMum
(·) is a scalar saturation function, enforcing the control input limitation,
defined similarly to what in (5.25), while uuc denotes the unconstrained control action
given by (B.1). In this way the windup of the integral term during the saturation period is
prevented, and it’s easy to verify that the overall control action calculated replacing I in
(B.1) with IAW in (B.2), will always lie inside the limits. However this simple strategy is
not suitable for high performance anti-windup, as required by modern formulation, indeed,
under some conditions, the above simple approach can lead to undesired system behaviors.
To motivate this claim consider the scenario when the proportional action alone, exceeds
171
Chapter B. SOME CONSIDERATIONS ON PRACTICAL PI ANTI-WINDUP SOLUTIONS
the saturation limit. In this case, performing strategy (B.2) could lead to reverse the
sign of the original unconstrained integral control action; e.g assume to have P, I > 0
and P > uM by (B.2) it follows IAW = uM − P < 0. Despite in practice this extreme
scenarios are rare, formally avoidance of such undesired behavior should be guaranteed,
as it can cause very sluggish or even unstable responses. A simple countermeasure is to
saturate also the proportional term before using it to compute IAW , hence strategy (B.2)
is modified as
IAW =
∫ tt0kix(τ)dτ if uuc ∈ [um, uM ]
satuMum
(uuc)− satuMum
(P ) otherwise(B.3)
and the overall control input is rewritten as
u(t) = satuMum
(P ) + IAW (B.4)
Such strategy ensures to always keep the coherence between the unconstrained and sat-
urated integral action, in the worst case scenario the integral reset to zero. In (B.3) the
priority is given to the proportional action, since all the available control effort is assigned
to it in case strong saturation conditions, causing P /∈ [um, uM ], take place. However,
depending on the specific applications, it can be profitable to preserve part of the control
authority for the integral action, for example if a partially known constant disturbance is
acting on the system. However it is easy to change the partitioning rule in (B.3), according
to a desired trade-off between the proportional and the integral action, it suffices to limit
the proportional term inside a set [u′m, u′M ] which is strictly contained inside the original
bounds [um, uM ].
The improved anti-windup scheme (B.3) poses an additional issue, when the proportional
and integral terms have opposite signs, adopting the saturated law (B.4) yields an un-
desired shed of the proportional action, especially if non symmetric saturation bounds
are considered. In this respect, a significant example is the wind turbine speed controller
proposed in 6; it has been remarked how a motoring behavior of the turbine should be
avoided, in this respect the lower bound for the control effort u defined by (6.13) and
(6.14) is set to zero. Now consider the case when the proportional action is negative since
the actual speed is following the reference, while the integral part, that can be thought as
an estimate of the mean aerodynamic torque, is positive. Applying (B.4) would lead to
reset the proportional term even if uuc = P + I ∈ [um, uM ], this is clearly unacceptable,
since the proportional term has a crucial stabilizing role as showed in 6.3.2.
These considerations can be clearly generalized to other class of systems, therefore, the
simple anti-windup strategy given by (B.3), (B.4) need to be refined considering the sign
of the unconstrained proportional and integral terms. A possible anti-windup solution,
preventing the undesired saturation of the proportional action when the overall control
effort lies inside the admissible region, and, at the same time, avoiding to reverse the sign
172
of the unconstrained integral part, is the following
IAW =
∫ tt0kix(τ)dτ if uuc ∈ [um, uM ]
satuMum
(uuc)− satuMum
(P ) if (uuc /∈ [um, uM ]) & sign(P ) = sign(I)
satuMum
(I) if (uuc /∈ [um, uM ]) & sign(P ) 6= sign(I)
(B.5)
u(t) = satuMum
(P + IAW ). (B.6)
Roughly speaking, when the proportional and the integral terms have opposite signs and
saturation occurs, the overall control effort, in the direction given by sign(I) is assigned
to the integral action, while the proportional term is left unchanged since it will steer the
sum P + IAW towards the opposite direction, and this overall action is eventually saturate
according to (B.6). While if the two control components have the same sign, the same
integral anti-windup strategy as in (B.3) is performed, so that the integral contribution is
reduced but the same direction as the ideally unconstrained one is maintained.
173
Bibliography
[1] N. Krikelis, “State feedback integral control with intelligent integrator,” Int. Journal
of Control, vol. 32, no. 3, pp. 465–473, 1980.
[2] K. J. Astrom and L. Rundqwist, “Integrator windup and how to avoid it,” Proc. of
IEEE American Control Conference, pp. 1693–1698, Pittsburg, PEN, USA, 1989.
[3] P. J. Campo, M. Morari, and C. N. Nett, “Multivariable anti-windup and bumpless
transfer: a general theory,” Proc. of IEEE American Control Conference, pp. 1712–
1718, Pittsburg, PEN, USA, 1989.
[4] R. Hanus, M. Kinnaert, and J. L. Henrotte, “Conditioning technique, a general
anti-windup and bumpless transfer method,” Automatica, vol. 23, no. 6, pp. 729–
739, 1987.
[5] R. Hanus, “Anti-windup and bumpless transfer: a survey,” Proc. of IEEE 12th
IMACS World Congress, pp. 56–65, Paris, France, 1988.
[6] R. Hanus and M. Kinnaert, “Control of constrained multivariable systems using the
conditioning technique,” Proc. of IEEE American Control Conference, pp. 1712–
1718, 1989.
[7] A. Casavola and E. Mosca, “Reference governor for constrained uncertain linear sys-
tems subject to bounded input disturbances,” Proc. of IEEE Conference on Decision
and Control, pp. 1117–1141, Kobe, Japan, 1996.
[8] E. G. Gilbert, I. Kolmansky, and K. T. Tan, “Discrete-time reference governors an
the nonlinear control of system with state and control constraints,” International
Robust Nonlinear Control, vol. 7, pp. 487–504, 1995.
[9] J. M. G. da Silva Jr. and S. Tarbouriesch, “Anti-windup design with guaranteed
region of stability: an lmi-based approach,” IEEE Tran. on Automatic Control,
vol. 50, no. 1, pp. 106–111, 2005.
[10] S. Miyamoto and G. Vinnicombe, “Robust control of plants with saturation nonlin-
earity based on coprime factor,” Proc. of IEEE Conference on Decision and Control,
pp. 2838–2840, Kobe, Japan, 1996.
174
Bibliography
[11] A. R. Teel and N. Kapoor, “The L2 anti-windup problem: Its definition and so-
lution,” Proc. of IEEE European Control Conference, pp. 1–6, Brussels, Belgium,
1997.
[12] L. Zaccarian and A. R. Teel, Modern Anti-windup Synthesis: control augmentation
for actuator saturation. Princeton University Press, Princeton (NJ), USA, 2011.
[13] S. Tarbouriech, J. M. G. da Silva Jr., and I. Queinnec, Stability and stabilization of
linear systems with saturating actuators. Springer-Verlag, London, UK, 2011.
[14] S. Galeani, S. Tarbouriech, M. C. Turner, and L. Zaccarian, “A tutorial on modern
anti-windup design,” European Journal of Control, vol. 15, no. 3-4, pp. 418–440,
2009.
[15] H. Khalil, Nonlinear systems 3rd edition. Prentice-Hall Inc. Upper Saddle River, NJ
,USA, 2002.
[16] C. Pittet, S. Tarbouriech, and C. Burgat, “Stability regions for linear systems with
saturating controls via circle and popov criteria,” IEEE Proc. of Conference on
Decision and Control, pp. 4518–4523, 1997.
[17] Z. Lin and T. Hu, Control Systems with Actuator Saturation. Birkhauser, Boston,
MA, USA, 1996.
[18] A. Saberi, Z. Lin, and A. R. Teel, “Control of linear systems with saturating actua-
tors,” IEEE Tran. on Automatic Control, vol. 41, no. 3, pp. 368–377, 1996.
[19] H. Fang, T. Hu, and Z. Lin, “Analysis of linear systems in the presence of actuator
saturation and L2-disturbances,” Automatica, vol. 40, pp. 1229–1238, 2004.
[20] T. Hu and Z. Lin, “Absolute stability with a generalized sector condition,” IEEE
Tran. on Automatic Control, vol. 49, no. 4, pp. 535–548, 2004.
[21] H. Hindi and S. Boyd, “Analysis of linear systems with saturation using convex
optimization,” Proc. of IEEE Conference on Decision and Control, pp. 903–908,
Florida, USA, 1998.
[22] Z. Lin and T. Hu, “An analysis and design method for linear systems subject to
actuator saturation and disturbance,” Automatica, vol. 38, no. 2, pp. 351–359, 2002.
[23] T. Hu, A. R. Teel, and L. Zaccarian, “Stability and performance for saturated sys-
tems via quadratic and nonquadratic lyapunov functions,” IEEE Tran. on Automatic
Control, vol. 51, no. 11, pp. 1170–1786, 2006.
[24] T. Hu, “Nonlinear control design for linear differential inclusions via convex hull
quadratic lyapunov functions,” Automatica, vol. 43, no. 1, pp. 685–692, 2007.
175
BIBLIOGRAPHY
[25] D. Dai, T. Hu, A. Teel, and L. Zaccarian, “Piecewise-quadratic lyapunov functions
for systems with dead zones or saturations,” System and Control Letters, vol. 58,
pp. 365–371, 2009.
[26] F. Blanchini and T. Hu, “Non-conservative matrix inequality conditions for sta-
bility/stabilizability of linear differential inclusions,” Automatica, vol. 46, no. 1,
pp. 190–196, 2010.
[27] S. Keerthi and E. G. Gilbert, “Optimal infinite horizon feedback control laws for
a general class of constrained discrete-time systems: stability and moving-horizon
approximations,” Journal of Opt. Theory, vol. 57, pp. 265–293, 1988.
[28] J. B. Rawlings and K. R. Muske, “The stability of constrained receding-horizon
control,” IEEE Tran. on Automatic Control, vol. 38, pp. 1512–1516, 1993.
[29] D. Limon, J. M. G. da Silva, T. Alamo, and E. F. Camacho, “Improved mpc design
based on saturating control laws,” European Journal of Control, vol. 11, pp. 112–122,
2005.
[30] A. Tilli and C. Conficoni, “Anti-windup scheme for current control of shunt ac-
tive filters,” Proc. of IEEE American Control Conference, pp. 6581–6587, Montreal,
Canada, 2012.
[31] L. Marconi, F. Ronchi, and A. Tilli, “Robust nonlinear control of shunt active filters
for harmonic current compensation,” Automatica, vol. 43, no. 2, pp. 252–263, 2007.
[32] A. Tilli, L. Marconi, and C. Conficoni, Recent Advances in Robust Control - Theory
and Applications in Robotics and Electromechanics, ch. Analysis, dimensioning and
robust control of Shunt Active Filter for harmonic currents compensation in electrical
mains. Intech Open Access Publisher, 2011.
[33] J. Hanschke, L. Marconi, and A. Tilli, “Averaging control of the dc-link voltage in
shunt active filters,” Proc. of IEEE Conference on Decision and Control, pp. 6211–
6216, San Diego, CA, USA, 2006.
[34] M. Johansson and A. Rantzer, “Computation of piecewise quadratic lyapunov func-
tions for hybrid systems,” IEEE Tran. on Automat. Contr, vol. 43, no. 4, pp. 555–
559, 1998.
[35] F. Blanchini and S. Miani, “A new class of universal lyapunov functions for the
control of uncertain linear sistems,” IEEE . Tran. on Automatic Control, vol. 44,
pp. 451–457, 1999.
[36] T. Hu and Z. Lin, “Composite quadratic lyapunov functions for constrained control
systems,” IEEE . Tran. on Automatic Control, vol. 48, pp. 440–450, 2003.
176
Bibliography
[37] H. Jung, C. Conficoni, A. Tilli, and T. Hu, “Modeling and control design for power
systems driven by battery/supercapacitor hybrid energy storage devices,” To appear
in Proc. of IEEE American Control Conference, Washington DC, USA, 2013.
[38] T. Burton, D. Sharpe, N. Jenkins, and E. Bossanyi, Wind energy handbook. Wiley
& Sons, Chichester, West Sussex (UK), 2001.
[39] A. Tilli and C. Conficoni, “Speed control for medium power wind turbines: an
integrated approach oriented to mppt,” Proc. of IFAC World Congress, pp. 544–
550, Milan, Italy, 2011.
[40] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems. Prentice
Hall,Englewood Cliffs, NJ, USA, 1989.
[41] F. D. Priscoli, L. Marconi, and A. Isidori, “Nonlinear observers as nonlinear internal
models,” Systems and Control Letters, vol. 55, pp. 640–649, 2006.
[42] H. K. Khalil, “High-gain observers in nonlinear feedback control,” Proc. of Int. Conf.
on Contr. Automation and Systems, pp. 232–243, Seul, Korea, 2008.
[43] A. Tilli and C. Conficoni, “Adaptive observer for three phase voltage source under
unbalanced conditions,” Proc. of IFAC Symposium on Nonlinear Control Systems,
pp. 1356–1361, Bologna, Italy, 2010.
[44] A. Tilli, G. Cignali, C. Rossi, C. Conficoni, and C. Rossi, “A synchronous coordinates
approach in position and speed estimation for permanent magnet synchronous ma-
chines,” Proc. of Mediterranean Control Conference, pp. 487–492, Barcelona, Spain,
2012.
[45] G. Grim, G. Hatfield, J. Postletwhaite, A. R. Teel, M. C. Turner, and L. Zaccarian,
“Antiwindup for stable linear systems with input saturation: an lmi based synthe-
sis,” IEEE Tran. on Automatic Control, vol. 48, no. 9, pp. 1509–1525, 2003.
[46] E. F. Mulder, M. Y. Kothare, and M. Morari, “Multivariable anti-windup controller
synthesis using linear matrix inequalities,” Automatica, vol. 37, no. 9, pp. 1407–1416,
2001.
[47] J. M. Biannic and S. Tarbouriech, “Optimization and implementation of dynamic
anti-windup compensators with multiple saturations in flight control systems,” Con-
trol Eng. Practice, vol. 17, pp. 703–713, 2009.
[48] J. M. G. da Silva Jr. and S. Tarbouriech, “Anti-windup design with guaranteed
regions of stability: an lmi-based approach,” Proc. of IEEE Conference on Decision
and Control, pp. 106–111, Hawaii, USA, 2003.
[49] C. Roos and J.-M. Biannic
177
BIBLIOGRAPHY
[50] A. Teel and N. Kapoor, “Uniting local and global controllers,” Proc. of IEEE Eu-
ropean Control Conference, pp. 1–6, Brussels, Belgium, 1997.
[51] A. R. Teel, “Anti-windup for exponentially unstable linear systems,” Int. J. on
Robust Nonlinear Control, vol. 9, pp. 701–716, 1999.
[52] S. Galeani, A. R. Teel, and L. Zaccarian, “Constructive nonlinear anti-windup design
for exponentially unstable linear plants,” Systems and Control Letters, vol. 56, no. 5,
pp. 357–365, 2007.
[53] F. Morabito, A. R. Teel, and L. Zaccarian, “Nonlinear anti-windup applied to euler-
lagrange systems,” IEEE Tran. on Robotics and Automation, vol. 20, no. 3, pp. 526–
537, 2004.
[54] A. Zheng, M. V. Kothare, and M. Morari, “Anti-windup design for internal model
control,” Int. Journal of Control, vol. 60, no. 5, pp. 1015–1024, 1994.
[55] L. Zaccarian and A. R. Teel, “A common framework for anti-windup, bumpless
transfer and reliable designs,” Automatica, vol. 38, no. 10, pp. 1735–1744, 2001.
[56] L. Zaccarian and A. R. Teel, “Nonlinear scheduled anti-windup design for linear
systems,” IEEE Tran. on Automatic Control, vol. 49, no. 11, pp. 2055–2061, 2004.
[57] E. G. Gilbert and K. T. Tan, “Linear systems with state and control constraints:
the theory and applications of maximal output admissible sets,” IEEE Tran. on
Automatic Control, vol. 36, pp. 1008–1020, 1991.
[58] E. G. Gilbert and K. T. Tan, “Linear systems with state and control constraints:
The theory and applications of maximal output admissible sets,” IEEE Tran. on
Automatic Control, vol. 36, pp. 1008–1020, 1991.
[59] A. Casavola and E. Mosca, “Robust command governors for constrained linear sys-
tems,” IEEE Tran. on Automatic Control, vol. 45, no. 11, pp. 2071–2077, 2000.
[60] C. governors for constrained nonlinear systems: direct nonlinear vs. lineariza-
tion based strategies Int. Journal of Robust and Nonlinear Control, vol. 9, pp. 677–
699, 1999.
[61] A. S. Morse, “Output controllability and systems synthesis,” SIAM Journal of Con-
trol, vol. 9, pp. 143–148, 1971.
[62] A. Isidori, Nonlinear control systems. Springer-Verlag, New York, NY, USA, 1994.
[63] L. Marconi, Tracking and regulation of linear and nonlinear systems. Phd disser-
tation, University of Bologna, Department of of Electronics, Computer Engineering
and Systems, 1998.
[64] L. Gyugy and E. Strycula, “Active ac power filters,” IEEE-IAS annual meeting,
pp. 529–535, Cincinnati, Ohio, USA, 1976.
178
Bibliography
[65] H. Akagi, “New trends in active filters for power conditioning,” IEEE Tran. on
Industrial Applications, vol. 32, pp. 1312–1332, 1996.
[66] B. Singh and K. Al-Haddad, “A review of active filters for power quality improve-
ment,” IEEE Tran. on Industrial Electronics, pp. 960–971, 1999.
[67] N. Mohan, T. Undeland, and W. P. Robbins, Power Electronics Converters, appli-
cations and design. Wiley & Sons, New York, USA, 1989.
[68] A. Chandra, B. Singh, and K. Al-Haddad, “An improved control algorithm of shunt
active filter for voltage regulation, harmonic elimination, power-factor correction
and balancing of nonlinear loads,” IEEE Tran. on Industrial Electronics, vol. 15,
pp. 495–507, 2000.
[69] S. G. Jeong and M. H. Woo, “Dsp-based active power filter with predictive current
control,” IEEE Tran. on Industrial Electronics, vol. 44, pp. 329–336, 1997.
[70] L. Marconi, F. Ronchi, and A. Tilli, “Robust perfect compensation of load harmonics
in shunt active filters,” Proc. of IEEE Conference on Decision and Control, pp. 2978–
2983, Paradise Island, Bahamas, 2004.
[71] F. Ronchi and A. Tilli, “Design methodology for shunt active filters,” EPE-PEMC,
Int. power elect. and motion contr. conference, 2002.
[72] P. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery. IEEE
Press, Piscataway, NY, USA, 1995.
[73] H. Akagi, Y. Kanagawa, and A. Nabae, “Instantaneous reactive power compensator
comprising switching devices without energy storage components,” IEEE Tran. on
Industry Application, vol. 20, no. 11, 1984.
[74] V. O. Nikiforov, “Adaptive non-linear tracking with complete compensation of un-
known disturbance,” European Journal of Control, vol. 41, no. 2, pp. 132–139, 1984.
[75] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in
System and Control Theory. SIAM Studies in Applied Mathematics, Philadelphia,
PEN, USA, 1994.
[76] A. Teel, L. Moreau, and D. Nesic, “A unified framework for input-to-state stability
in systems with two time scales,” IEEE Tran. on Automatic Control, vol. 48, no. 9,
pp. 1526–1544, 2003.
[77] S. Sanders, M. Noworolski, X. Liu, and G. Verghese, “Generalized averaging method
for power conversion circuits,” IEEE Tran. on Power Electronics, vol. 6, no. 2,
pp. 251–259, 1991.
179
BIBLIOGRAPHY
[78] E. B. Castelan, I. Queinnec, S. Tarbouriech, and J. M. G. da Silva Jr., “Lmi ap-
proach for L2-control of linear systems with saturating actuators,” Proc. of IFAC
Symposium on Nonlinear Control Systems, pp. 287–292, Stuttgart, Germany, 2004.
[79] T. Hu, A. R. Teel, and Z. Lin, “Lyapunov characterization of forced oscillations,”
Automatica, vol. 41, no. 7, pp. 1723–1735, 2005.
[80] F. Blanchini and A. Megreski, “Robust state feedback control of ltv systems: non-
linear is better than linear,” IEEE Tran. on Automatic Control, vol. 44, no. 4,
pp. 802–807, 1999.
[81] A. Hassibi, J. How, and S. Boyd, “A path-following method for solving bmi problems
in control,” Proc. of IEEE American Control Conference, pp. 1385–1389, 1999.
[82] K. Goh, M. Safonov, and G. Papavassilopoulos, “A global optimization approach for
the bmi problem,” Proc. of IEEE Conference on Decision and Control, pp. 325–331,
Lake Buena Vista, FL, USA, 1994.
[83] A. Rantzer and M. Johansson, “Piecewise linear quadratic optimal control,” IEEE
Tran. on Automat. Contr, vol. 45, no. 4, pp. 629–637, 2000.
[84] F. Blanchini and S. Miani, “Nonquadratic lyapunov functions for robust control,”
Automatica, vol. 31, no. 3, pp. 451–561, 1995.
[85] W. P. Dayawansa and C. F. Martin, “A converse lyapunov theorem for a class of
dynamical systems which undergo switching,” IEEE Tran. on Automatic Control,
vol. 44, no. 4, pp. 751–760, 1999.
[86] R. K. Brayton and C. H. Tong, “Constructive stability and asymptotic stability of
dynamical systems,” IEEE Tran. on Circuits and Systems, vol. 27, no. 11, pp. 1121–
1130, 1980.
[87] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Homogeneous lyapunov functions for
systems with structured uncertainties,” Automatica, vol. 39, no. 6, pp. 1027–1035,
2003.
[88] R. Goebel, T. Hu, and A. R. Teel, “Dual matrix inequalities in stability and perfor-
mance analysis of linear differential/difference inclusions,” Current Trends in Non-
linear Systems and Control, Boston, MA, 2006.
[89] R. Goebel, A. R. Teel, T. Hu, and Z. Lin, “Conjugate convex lyapunov functions
for dual linear differential inclusions,” IEEE Tran. on Automatic Control, vol. 51,
no. 4, pp. 660–666, 2006.
[90] T. Hu and Z. Lin, “Properties of convex hull quadratic lyapunov functions,” IEEE
Tran. on Automatic Control, vol. 49, no. 4, pp. 1162–1167, 2004.
180
Bibliography
[91] S. Boyd and L. Vandeberghe, Convex optimization. Cambridge University Press,
Cambridge, UK, 2004.
[92] T. Hu, R. Goebel, and A. Teel, “Conjugate lyapunov functions for saturated linear
systems,” Automatica, vol. 41, pp. 1949–1956, 2005.
[93] M. B. Camara, H. Gualous, F. Gustin, and A. Berthon, “Design and new control
of dc/dc converters to share energy between supercapacitors and batteries in hybrid
vehicles,” IEEE Tran. on Vehicular Technologies, vol. 57, no. 5, pp. 2721–2735, 2008.
[94] M. B. Camara, H. Gualous, F. Gustin, A. Berthon, and B. Dakyo, “Dc/dc converter
design for supercapacitor and battery power management in hybrid vehicle applica-
tions with polynomial control strategy,” IEEE Tran. on Industr. Electronics, vol. 57,
no. 3, pp. 587–597, 2010.
[95] J. Cao and A. Emadi, “A new battery/ultracapacitor hybrid energy storage system
for electric, hybrid, and plug-in hybrid electric vehicles,” IEEE Tran. on Power
Electronics, vol. 27, no. 1, pp. 122–132, 2012.
[96] S. M. Lukic, S. G. Wirasingha, F. Rodriguez, and A. E. J. Cao, “Power management
of an ultracapacitor/battery hybrid energy storage system in an hev,” Proc. of IEEE
Vehicle Power and Propulsion Conference, pp. 1–6, 2006.
[97] L. Wei, G. Joos, and J. Belanger, “Real-time simulation of a wind turbine genera-
tor coupled with a battery supercapacitor energy storage system,” IEEE Tran. on
Industrial Electronics, vol. 57, no. 4, pp. 1137–1145, 2010.
[98] W. Li and G. Joos, “A power electronic interface for a battery supercapacitor hybrid
energy storage system for wind applications,” Proc. of IEEE Power Electron. Spec.
Conf., pp. 1762–1768, 2008.
[99] M. Glavin, P. K. W. Chan, S. Armstrong, and W. G. Hurley, “A stand-alone photo-
voltaic supercapacitor battery hybrid energy storage system,” Proc. of IEEE Power
Electron. and Motion Contr. Conf., pp. 1688–1695, 2008.
[100] H. Zhou, T. Bhattacharya, D. Tran, T. S. T. Siew, and A. M. Khambadkone, “Com-
posite energy storage system involving battery and ultracapacitor with dynamic
energy management in microgrid applications,” IEEE Tran. on Power Electronics,
vol. 26, no. 3, pp. 923–930, 2011.
[101] S. Pay and Y. Baghzouz, “Effectiveness of battery-supercapacitor combination in
electric vehicles,” Proc. of IEEE Conf. on Power Technologies, pp. 75–81, Bologna,
Italy, 2003.
[102] M. Choi, S. Kim, and S. W. Seo, “Energy management optimization in a bat-
tery/supercapacitor hybrid energy storage system,” IEEE Tran. on Smart Grid,
vol. 3, no. 1, pp. 463–472, 2012.
181
BIBLIOGRAPHY
[103] L. Gao, R. A. Dougal, and S. Liu, “Power enhancement of an actively controlled
battery/ultracapacitor hybrid,” IEEE Tran. on Power Electronics, vol. 20, no. 1,
pp. 236–243, 2005.
[104] D. L. Cheng and M. Wismer, “Active control of power sharing in a bat-
tery/ultracapacitor hybrid source,” IEEE Conf. on Industr. Electronics and Ap-
plications, pp. 2913–2918, 2007.
[105] M. Camara, H. Gualous, B. Dakyo, C. Nichita, and P. Makany, “Buck-boost con-
verters design for ultracapacitors and lithium battery mixing in hybrid vehicle ap-
plications,” Proc. of IEEE Conf. on Vehicle Power and Propulsion, pp. 1–6, 2010.
[106] Y. Zhang, Z. Jiang, and X. Yu, “Control strategies for battery/supercapacitor hybrid
energy storage systems,” Proc. of IEEE Energy2030 Conference, pp. 1–6, 2008.
[107] T. Hu, “A nonlinear system approach to analysis and design of power electronic
converters with saturation and bilinear terms,” IEEE Tran. on Power Electronics,
vol. 2, no. 26, 2011.
[108] R. D. Middlebrook, “A general unified approach to modeling switching-converter
power stages,” Proc. of IEEE Power Electronics Specialists Conf., 1976.
[109] Y. Yao, F. Fassinou, and T. Hu, “Stability and robust regulation of battery driven
boost converter with simple feedback,” IEEE Tran. on Power Electronics, vol. 26,
no. 9, pp. 2614–2626, 2011.
[110] S. Banerjee and G. C. Verghese, Nonlinear Phenomena in Power Electronics: Bi-
furcations, Chaos, Control, and Applications. Wiley & Sons, USA, 2001.
[111] W. E. Leithead and P. Connor, “Control of variable speed wind turbines: design
task,” IEEE Int. Journal of Control, vol. 73, no. 14, pp. 2614–2626, 2000.
[112] F. D. Bianchi, R. J. Mantz, and C. F. Christiansen, “Gain scheduling control of
variable-speed wind energy conversion systems using quasi-lpv models,” Control En-
gineering practice, vol. 13, pp. 247–255, 2005.
[113] R. Rocha, L. S. M. Filho, and M. V. Bortolu, “Optimal multivariable control for
wind energy conversion system a comparisono between h2 and h∞ controllers,” Proc.
of IEEE Conference on Decision and Control”, Seville, Spain, pp. 7906–7911, 2005.
[114] C. Sloth, T. Esbensen, M. Niss, J. Storustrup, and P. F. Odgaard, “Robust lmi-
based control of wind turbines with parametric uncertainties,” Proc. of IEEE Multi-
conference on systems and control, pp. 776–781, 2009.
[115] C. L. Bottasso, A. Croce, and B. Savini, “Performance comparison of control schemes
for variable-speed wind turbines,” Journal of physics: Conference series 75, 2007.
182
Bibliography
[116] R. Datta and V. T. Ranghanathan, “A method of tracking the peak power points for
a variable speed wind energy conversion system,” IEEE Tran. on energy conversion,
vol. 18, no. 10, pp. 163–168, 2003.
[117] E. Koutrolis and K. Kalaitzakis, “Design of a maximum power tracking system for
wind-energy-conversion applications,” IEEE Tran. on Industrial Electronics, vol. 53,
no. 2, pp. 486–491, 2006.
[118] Q. Wang and L. Chang, “An intelligent maximum power extraction algorithm for
inverter-based variable speed wind turbine systems,” IEEE Tran. on Power Elec-
tronics, vol. 19, no. 5, pp. 1242–1249, 2004.
[119] K. E. Johnson, L. Y. Pao, M. J. Balas, V. Kulkarni, and L. J. Fingersh, “Stability
analysis of an adaptive torque controller for variable speed wind turbines,” Proc.
of IEEE Conference on Decision and Control, pp. 4087–4094, Paradise Island, Ba-
hamas, 2004.
[120] T. Ackermann, Wind power in power systems. Wiley & Sons, Chichester, West
Sussex, England, 2005.
[121] R. C. Dugan, M. F. M. Granaghan, S. Santoso, and H. W. Beaty, Electrical power
system quality. Mc Graw-Hill, 2004.
[122] M. H. J. Bollen, G. Yalcinkaya, and P. A. Crossley, “Characterization of voltage sags
in industrial distribution systems,” IEEE Tran. on Industry Applications, no. 34,
pp. 682–688, 1998.
[123] P.Pillay and M. Manyage, “Definitions of voltage unbalance,” IEEE Power Engi-
neering Review, pp. 50–52, 2001.
[124] C. EN-50160:, “Voltage characteristics of electricity, supplied by public distribution
systems,” Brussels, 1994.
[125] E. N. Gmbh, “Grid code, high and extra high voltage,” Bayeruth, 2006.
[126] EirGrid, “Irish grid code,” version 3.4, 2009.
[127] H. S. Song and K. Nam, “Instantaneus phase-angle estimation algorithm under
unbalanced voltage sag conditions,” Proc. of IEEE Generation, Transmission and
Distribution, vol. 147, no. 6, pp. 409–415, 2000.
[128] H. S. Song and I. Joo, “Source voltage sensorless estimation scheme for pwm rectifiers
under unbalanced conditions,” IEEE Tran. on Industrial Electronics, vol. 50, no. 6,
pp. 1238–1245, 2003.
[129] P. Vas, Sensorless Vector and Direct Torque Control. Oxford, New York, NY, USA,
1998.
183
BIBLIOGRAPHY
[130] R. Marino, P. Tomei, and C. M. Verrelli, Induction Motor Control Design. Springer-
Verlag, London, UK, 2010.
[131] S. Ichikawa, M. Tomita, S. Doki, and S. Okuma, “Sensorless control of pmsm using
on-line parameter identification based on systems identification theory,” IEEE Tran.
on Industrial Electronics, vol. 53, no. 2, pp. 363–373, 2003.
[132] F. Poulain, L. Praly, and R. Ortega, “An observer for pmsm with application to
sensorless control,” Proc. of IEEE Conference on Decision and Control, pp. 5390–
5395, Cancun, Mexico, 2008.
[133] S. Ibarra, J. Moreno, and G. Espinosa-Perez, “Global observability analysis of sen-
sorless induction motor,” Automatica, vol. 40, no. 2, pp. 1079–1085, 2004.
[134] A. Bado, S. Bolognani, and M. Zigliotto, “Effective estimation of speed and rotor
position of a pm synchronous motor drive by a kalman filtering technique,” Proc. of
IEEE Power Electronics Specialists Conference, pp. 951–957, Toledo, Spain, 1992.
[135] J. Solsona, M. I. Valla, and C. Muravchik, “A nonlinear reduced order observer for
pmsm,” IEEE Tran. on Industrial Electronics, vol. 43, no. 4, pp. 492–497, 1996.
[136] N. Matsui, “Sensorless pm brushless dc motor drives,” IEEE Tran. on Industrial
Electronics, vol. 43, no. 2, pp. 300–308, 2010.
[137] B. N. Mobarakeh, F. Meibody-Tabar, and F. Sargos, “Robustness study of a model-
based technique for mechanical sensorless control of pmsm,” Proc. of IEEE Power
Electronics Specialists Conference, pp. 811–816, Vancouver, Canada, 2001.
[138] R. Ortega, L. Praly, A. Astolfi, J. Lee, and K. Nam, “Estimation of rotor position
and speed of permanent magnet synchronous motors with guaranteed stability,”
IEEE Tran. on Control Systems Technology, vol. 19, no. 3, pp. 601–614, 2010.
[139] N. Kazantzis and C. Kravaris, “Nonlinear observer design using lyapunov auxiliary
theorem,” Systems and Control Letters, vol. 34, pp. 241–247, 1998.
[140] A. Astolfi, D. Karagiannis, and R. Ortega, Nonlinear and Adaptive Control with
Applications. Springer-Verlag, Berlin, Germany, 2007.
[141] S. Tarbouriech, C. Prieur, and J. M. G. da Silva Jr., “Stability analysis and stabiliza-
tion of systems presenting nested saturations,” IEEE Tran. on Automatic Control,
vol. 51, no. 8, pp. 1364–1371, 2006.
184